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CARNEGIE INSTITUTE OF TECHNOLOGY LIBRARY PRESENTED BY Dr*Lloyd L. Dines A COURSE IN MATHEMATICAL ANALYSIS FUNCTIONS OF A COMPLEX VARIABLE BEING PART I OF VOLUME II BY EDOUARD GOURSAT PROFESSOR OF MATHEMATICS, THE UNIVERSITY OF PARIS TRANSLATED BY EARLE RAYMOND HEDRICK PROFESSOR OF MATHETtfATICS, THE UNIVERSITY OF MISSOURI AND OTTO DUNKEL ASSOCIATE PROFESSOR OF MATHEMATICS, WASHINGTON UNIVERSITY GINN AND COMPANY BOSTON NEW YORK - CHICAGO LONDON ATLANTA DALLAS COLUMBUS SAN FRANCISCO COPYRIGHT, 1916, BY EARLE RAYMOND HEDRICK AND OTTO DUNKEL ALL BIGHTS RESERVED PRINTED IN THE UNITED STATES OF AMERICA 226,11 fltfttnaum GINN AND COMPANY . PRO PRIETORS BOSTON USA AUTHOR'S PREFACE SECOND FRENCH EDITION The first part of this volume has undeigone only slight changes, while the lather important modifications that have been made appear only m the last chapters In the first edition I was able to devote but a few pages to par- tial differential equations of the second ordei and to the calculus of variations In ordei to present in a less summary manner such broad subjects, I have concluded to defer them to a third volume, which will contain also a sketch of the recent theory of integral equations The suppression of the last chapter has enabled me to make some additions, of which the most important relate to linear differential equations and to partial differential equations of the first Oldei E GOTJESAT TRANSLATORS' PREFACE As the title indicates, the present volume is a translation of the first half of the second volume of Goursat's "Cours d' Analyse " The decision to publish the translation in two parts is due to the evi- dent adaptation of these two portions to the introductory courses in American colleges and universities in the theory of functions and in differential equations, respectively. After the cordial reception given to the translation of Goursat's first volume, the continuation was assured. That it has been delayed so long was due, in the first instance., to our desire to await the appearance of the second edition of the second volume in French The advantage in doing so will be obvious to those who have observed the radical changes made in the second (French) edition of the second volume Volume I was not altered so radi- cally, so that the present English translation of that volume may be used conveniently as a companion to this ; but references are given here to both editions of the first volume, to avoid any possible difficulty in this connection. Our thanks are due to Professor Goursat, who has kindly given us his permission to make this translation, and has approved of the plan of publication in two parts He has also seen all proofs in English and has approved a few minor alterations made in transla- tion as well as the translators' notes. The responsibility for the latter rests, however, with the translators. E. R. HEDRICK OTTO DTJNKEL CONTENTS PAGE CHAPTER I ELEMENTS OF THE THEORY . 3 I GENERAL PRINCIPLES ANALYTIC FUNCTIONS 3 1 Definitions . . 3 2 Continuous functions of a complex variable 6 3 Analytic functions . . 7 4 Functions analytic throughout a region . 11 5 Rational functions . .12 6 Ceitam irrational functions . 13 7 Single-valued and multiple-valued functions 17 II POWER SERIES WITH COMPLEX TERMS ELEMENTARY TRANSCENDENTAL FUNCTIONS 18 8 Circle of conveigence , . 18 9 Double seiies . . 21 10 Development of an infinite pioduct in power series 22 11 The exponential function . 23 12. Trigonometric functions . . 26 13 Logarithms . . 28 14 Inverse functions arc sin z, arc tan s 30 15 Application to the integral calculus . 33 16 Decomposition of a rational function of sin z and cos z into simple elements . . 35 17. Expansion of Log (1 + z) . .38 18 Extension of the binomial formula . .40 III CONFORMAL REPRESENTATION . . 42 19 Geometric interpretation of the derivative . 42 20. Conformal transformations in general .... 45 21. Conformal representation of one plane on another plane . . 48 22 Riemann's theorem ... 50 23 Geographic maps , . .52 24 Isothermal curves .... . . 54 EXERCISES 56 Viii CONTENTS PAGE CHAPTER II THE GENERAL THEORY OF ANALYTIC FUNC- TIONS ACCORDING TO CAUCHY ... 60 I. DEFINITE INTEGRALS TAKEN BETWEEN IMAGINARY LIMITS 60 25 Definitions and general principles . 60 26 Change of variables 62 27 The formulae of Weierstrass and Daiboux 64 28 Integrals taken along a closed cuive . 66 31 Generalization of the formulae of the integral calculus , 72 32 Another proof of the preceding results . . 74 II CAUCHY'S INTEGRAL TAYLOR'S AND LAURENT'S SERIES SINGULAR POINTS EESIDUES . 75 33 The fundamental formula 75 34 Morera's theorem . 78 35 Taylor's series 78 36 Liouville's theorem 81 37 Laurent's series . 81 38 Other series . .84 39 Series of analytic functions 86 40 Poles ... . ,88 41 Functions analytic except for poles . 90 42 Essentially singular points . 01 43 Residues . 04 III. APPLICATIONS OF THE GENERAL THEOREMS , 05 44 Introductory remarks .... . 05 45 Evaluation of elementary definite integrals . 06 46 Various definite integrals ... .07 47 Evaluation of T( T(l-^) . . 100 48 Application to functions analytic except for poles 101 49 Application to the theory of equations ... . 103 50 Jensen's formula . .... 104 51. Lagrange's formula . ... 106 52 Study of functions for infinite values of the variable 109 IV. PERIODS OF DEFINITE INTEGRALS . . ... 112 53 Polar periods . . . . Hg 54. A study of the integral J^dz/^/l z* 114 55. Periods of hyperelliptic integrals . .... 116 56 Periods of elliptic integrals of the first kind 120 EXERCISES 122 CONTENTS ix PAGE CHAPTER III SINGLE-VALUED ANALYTIC FUNCTIONS . 127 L WEIERSTRASS'S PRIMARY FUNCTIONS. MITTAG-LEFFLER'S THEOREM . 127 57 Expression of an integial function as a product of primary functions 127 58 The class of an integial function . 132 59 Single-valued analytic functions with a finite number of singular points . . . 132 60. Single-valued analytic functions with an infinite number of singular points . . .... 134 61. Mittag-Leffler's theorem . . . 134 62. Certain special cases . . 137 63 Cauchy's method . . 139 64. Expansion of ctn x and of sin x . 142 II. DOUBLY PERIODIC FUNCTIONS ELLIPTIC FUNCTIONS 145 65 Periodic functions Expansion in series 145 66 Impossibility of a single-valued analytic function with three periods . . . 147 67 Doubly periodic functions . . 149 68 Elliptic functions General properties . . 150 69 The fimctionp(w) . . . . 154 70 The algebraic relation between p(u) and p'(u) . 158 71 The function (u) . . 159 72 The f unction <r(w) . . .162 73 General expressions for elliptic functions . 163 74 Addition formulae . 166 75. Integration of elliptic functions . . 168 76 The function . 170 III. INVERSE FUNCTIONS. CURVES OF .DEFICIENCY ONE 172 77. Relations between the penods and the invariants . . 172 78 The inverse function to the elliptic integral of the first kind 174 79. A new definition of p (u) by means of the invariants . . . 182 80 Application to cubics in a plane . . . . 184 81. General formulae for parameter representation . 187 82 Curves of deficiency one .... . . 191 EXERCISES . . . .... . .193 CHAPTER IV. ANALYTIC EXTENSION 196 L DEFINITION OF AN ANALYTIC FUNCTION BY MEANS OF ONE OF ITS ELEMENTS 196 83 Introduction to analytic extension 196 84. New definition of analytic functions 199 x CONTENTS PAGE 85 Singular points . . . 204 86 Geneial pioblem . . 206 II NATURAL BOUNDARIES CUTS 208 87 Smgulai lines Natuial boundanes 208 88 Examples 211 89 Singularities of analytical expiessions 213 90 Heinute's foimula . 215 EXERCISES . . 217 CHAPTER V ANALYTIC FUNCTIONS OF SEVERAL VARIABLES 219 I GENERAL PROPERTIES 219 91 Definitions . 219 92 Associated cucles of convergence 220 93 Double integrals 222 94. Extension ot Cauchy's theoiems 225 95 Functions repiesentod by definite integrals . 227 96 Application to the F function . 229 97 Analytic extension of a function of two variables 231 II. IMPLICIT FUNCTIONS A.LGEBRAIC FUNCTIONS 232 98 Weiersti ass's tlieoiem 232 99 Critical points . . .236 100 Algebraic functions . , 240 101 Abehan integrals . . 243 102 Abel's theoiem . 244 103 Application to hyper elliptic integrals . 247 104 Extension of Lag range's formula . 250 EXERCISES , 2 r >2 INDEX . 253 A COURSE IN MATHEMATICAL ANALYSIS VOLUME II. PART I THEOEY OF FUNCTIONS OF A COMPLEX VARIABLE CHAPTER I ELEMENTS OF THE THEORY I GENERAL PRINCIPLES ANALYTIC FUNCTIONS 1. Definitions. An ^ma,g^nary quantity, or complex quantity ', is any expression of the foim a + bi where a and b are any two real num- bers whatever and i is a special symbol which has been introduced in order to generalize algebra. Essentially a complex quantity is nothing but a system of two real numbers arranged in a certain order Although such expressions as a -h bi have in themselves no concrete meaning whatever, we agree to apply to them the ordinary rules of algebra, with the additional convention that i* shall be replaced throughout by 1 Two complex quantities a -f- bi and a' + b'i are said to be equal if a = a' and b = b' The sum of two complex quantities a + bi and c + di is a symbol of the same form a 4- o + (b + d)t, the differ- ence a + bi (0-f di) is equal to a c + (b d)L To find the product of a + bi and c + di we carry out the multiplication accord- ing to the usual rules for algebraic multiplication, replacing i z by 1, obtaining thus (a 4. H}(c + di) = ac - bd+(ad + be)i. The quotient obtained by the division of a + bi by c + di is defined to be a third imaginary symbol x + yi, such that when it is multiplied by c + di, the product is a + bi The equality a -f bi = (c + di) (x + yf) is equivalent, according to the rules of multiplication, to the two relations cx whence we obtain 4 ELEMENTS OF THE THEORY [I, 1 The quotient obtained by the division of a -j- In by c + di is repre- sented by the usual notation for fi actions in algebra, thus, a + bi A convenient way of calculating x and ?/ is to multiply numerator and denominator of the fraction by c d i and to develop the indicated products All the properties of the fundamental operations of algebra can be shown to apply to the operations carried out on these imaginary sym- bols Thus, if A, B, C, denote complex numbers, we shall have ,4 B=B A, A B C=J (B C), A (B + C) = AB + AC, and so on. The two complex quantities a + bi and a bi aie said to be conjugate imaginanes The two complex quantities a + bi and a bi, whose sum is zeio, aie said to be negatives of each other or symmetric to each othei Given the usual system of lectangular axes in a plane, the complex quantity a + bi is lepresented by the point M of the plane xOy, whose cooidmates are x = a and y = b In this way a concrete representa- tion is given to these puiely symbolic expressions, and to eveiy proposition established for complex quantities there is a correspond- ing theorem of plane geometry But the greatest advantages resulting from this representation will appear later Real numbers correspond to points on the ct-axis, which for this reason is also called the aaib of reals Two conjugate imagmaries a + bi and a II correspond to two points symmetrically situated with respect to the o?-axis Two quantities a + bi and a bl are lepiesented by a pair of points symmetric with respect to the ongin The quantity a + In, which corresponds to the point M with the coordinates (a, ft), is sometimes called its affix.* When there is no danger of ambiguity, we shall denote by the same letter a complex quantity and the point which represents it Let us ;jom the ongin to the point M with coordinates (a, b) by a segment of a straight line The distance OM is called the absolute value of a + bij and the angle through which a ray must be turned from Ox to bring it in coincidence with OM (the angle being measured, as in trigonometry, from Ox toward Oy) is called the angle of a + bi * This term is not much used m English, but the French frequently use the coire- sponding word affixe TRANS GENERAL PRINCIPLES Let p and. a> denote, respectively, the absolute value and the angle of a -f fa > between the real quantities <z, I, p, o> there exist the two rela- tions a = p cos a), Z> = p sin <o, whence we have cos <o = sin a? = The absolute value p, which is an essentially positive number, is determined without ambiguity , whereas the angle, being given only by means of its trigonometric functions, is deteimined except for an additive multiple of 2 TT, which was evident from the definition itself. Hence every complex quantity may have an infinite number of angles, forming an arithmetic progression in which the successive terms differ by 2 TT. In order that two complex quantities be equal, their absolute values must be equal, and moieovei their angles must differ only by a multiple of 2 TT, and these conditions are sufficient. The absolute value of a complex quantity z is represented by the same symbol \&\ which is used for the absolute value of a real Quantity Let # = a + bi, &' = a* -\- b'i be two complex numbers and rn, m' the corresponding points , the sum & -f ' is then represented by the point m", the vertex of the parallelogram constructed upon Om, Om f . The three sides of the tuangle Om m n (Fig 1) are equal respectively to the absolute values of the quantities , z 1 , & -f- 2' From this we conclude that the absolute value of the sum of two quanti- ties u less than or at most equal to the sum of the absolute values of the two quantities, and greater than or at least equal to their difference Since two quantities that are negatives of each, other have the same absolute value, the theorem is also true for the absolute value of a difference Finally, we see in the same way that the absolute value of the sum of any number of complex quantities is at most equal to the sum of their absolute values, the equality holding only when all the points representing the different quantities are on the same ray starting from the origin. If through the point m we draw the two straight lines mx r and my' parallel to Ox and to Oy, the coordinates of the point m' in this system of axes will be a' a and b' b (Fig 2). The point m 1 then represents # f # in the new system , the absolute value of x FIG. 1 y 6 ELEMENTS OF THE THEORY [I, 1 #' & is equal to the length mm', and the angle of ' 2 is equal to the angle which the direction mm' makes with mx' Draw thiough a segment Oi^ equal and par- allel to mm' , the extremity m 1 of this segment represents f & in the system of axes Ox, Oy But the figuie Oni'vi^ is a parallelo- gram , the point m 1 is therefore the symmetric point to m with respect to c, the middle point of Om' _!_ J.VT 4 Finally, let us obtain the for- mula which gives the absolute value and angle of the product of any number of factors Let z k = pi (cos <DJL 4~ * sin o> A ), (7c = 1, 2, , ri), be the factors , the rules for multiplication, together with the addi- tion f orrnulse of trigonometry, give for the product 4- i sm (^ 4 <t> 2 + ' + 'Ol which shows that the absolute value of a product is equal to the product of the absolute values, and the angle of a product is equal to the siim of the angles of the factors From this follows very easily the well-known formula of Be Moivre cos m<t) 4- i sin w = (cos <o 4 * sin >) m } which contains in a very condensed form all the trigonometric for- mulse for the multiplication of angles The introduction of imaginary symbols has given complete gener- ality and symmetry to the theory of algebraic equations It was in the treatment of equations of only the second degree that such ex- pressions appeared for the first time Complex quantities are equally important in analysis, and we shall now state precisely what mean- ing is to be attached to the expression a function of a complex variable. 2. Continuous functions of a complex variable. A complex quantity z =s x 4- yi") where x and y are two real and independent variables, is a complex variable If we give to the word function its most general meaning, it would be natural to say that every other complex quantity u whose value depends upon that of is a function of . I, 3] GENERAL PRINCIPLES 7 Certain familiar definitions can be extended directly to tliese func- tions Thus, we shall say that a function u =f(z) is continuous if the absolute value of the difference f(& + &)/() approaches zeio when the absolute value of h approaches zero, that is, if to every positive number we can assign another positive number TJ such that |/(* + A) -/(*)!< provided that | Ji \ be less than 77 Asenes > (*) + 1 (*)+ +.(*)+ , whose terms are functions of the complex vaiiable # is uniformly convergent in a region A of the plane if to eveiy positive number e we can assign a positive integer N such that for all the values of & in the region A, provided that n ^ N It can be shown as before (Vol I, 31, 2d ed , 173, 1st ed ) that if a series is uniformly convergent in a region A, and if each of its terms is a continuous function of 111 that region, its sum is itself a continuous function of the variable & in the same region. Again, a series is uniformly convergent if, for all the values of consideied, the absolute value of each term ^| 1S ^ ess tnan tne corresponding term ? H of a convergent series of real positive con- stants The series is then both absolutely and umfoimly convergent. Every continuous function of the complex variable is of the form u = P(x 9 2/) + Q(, y)i, where P and Q are real continuous functions of the two real variables oe, y~ If we were to impose no other restrictions, the study of functions of a complex variable would amount simply to a study of a pair of functions of two real variables, and the use of the symbol i would introduce only illusory simplifications In order to make the theory of functions of a com- plex vaiiable present some analogy with the theoiy of functions of a real variable, we shall adopt the methods of Cauchy to find the con- ditions which the functions P and Q, must satisfy in order that the expression P + Qi shall possess the fundamental properties of func- tions of a real variable to which the processes of the calculus apply 3. Analytic functions. If /() is a function of a real variable x whach has a derivative, the quotient h 8 ELEMENTS OF THE THEORY [I, 3 approaches f(x) when 7i approaches zero. Let us determine in the same way under what conditions the quotient AM A/> + &AQ As: AT/ 4- will approach a definite limit when the absolute value of A# approaches zero, that is, when Aa? and Ay approach zero independently. It is easy to see that this will not be the case if the functions P(x } y) and Q(x, y) are any functions whatever, for the limit of the quotient Aw/A# depends in general on the latio Ay/Ax, that is, on the way in which the point repiesenting the value of -f- h approaches the point representing the value of z Let us first suppose y constant, and let us give to x a value x -f- A# differing but slightly from x , then Att_P(s + As,y)-.P(ar,y) | ^ Q (x + Ax, y) - Q (x, y) A# Ax Ace In order that this quotient have a limit, it is necessary that the functions P and Q possess partial derivatives with respect to x, and in that case Att 8/> 8fl lim = -M . As <?aj cos Next suppose x constant, and let us give to y the value y + A?/ , we have y + Ay) - P(x, y) ^ <a(ag, ?/ + A?/) - Q(g, y) Ay and in this case the quotient will have for its limit dQ .dP 5 -- 1-%- dy dy if the functions P and Q possess partial derivatives with respect to y In order that the limit of the quotient be the same in the two cases, it is necessary that a/>_<2 ^!__Q ^ ' dx dy dy"" dx Suppose that the functions P and Q satisfy these conditions, and that the partial derivatives dP/dx, dP/dy, dQ/dx, dQ/dy are con- tinuous functions. If we give to x and y any increments whatever, Aaj, Ay, we can write AP = P(x + Aa, y + &y)-P(x + bx,y) + P(x + Aa?, y)- P(x,y) (x + Ax, y + 0Ay) + Aa;P; (as + 0'Aa, y) I, 3] GENERAL PRINCIPLES 9 where and 0' are positive numbers less than unity , and in the same way AQ = A[<(aj, y) + e'] + Ay [<&(, y) + <], where 6, e f , 1? cj approach zero with Ax and Ay The difference Aw = AP 4- *AQ can be written by means of the conditions (1) in the form, where 17 and 17' are infinitesimals We have, then, *!*=:! 4. dQ Ace If 1 17 1 and 1 77'! are smaller than a number a, the absolute value of the complementary term is less than 2 a This term will therefore ap- proach zero when Ax and Ay approach zero, and we shall have . A oP , .dQ lm "fe + ^- The conditions (1) are then necessary and sufficient in order that the quotient Aw/A# have a unique limit for each value of , provided that the partial derivatives of the functions P and Q be continuous The function u is then said to be an analytic function * of the variable z, and if we represent it by f(z), the derivative / f () is equal to any one of the following equivalent expressions : /ON ^/x dp , 3 30* - dp dp dp a 3. .dQ ( 2 ) /W^T" + *T" "a -- *"fl" !B3 1 -- '^ = T~ + *^~" x x '.^ y &c ^cc ^y 3y ^a? oy oy ex It is important to notice that neither of the pair of functions P(x, y), Q(x, y) can be taken arbitrarily. In fact, if P and Q have derivatives of the second order, and if we differentiate the first of the relations (1) with lespect to x, and the second with respect to y, we have, adding the two resulting equations, * Cauchy made frequent use of the term monogene, the equivalent of which, mono- gentc, is sometimes used in English The term synectique is also sometimes used in French We shall use by preference the term analytic, and it will be shown latei that this definition agrees with the one which has already been given (I, 197, 2d ed., 191, 1st ed) 10 ELEMENTS OF THE THEORY [I, 3 We can show in the same way that AQ = The two functions p(x, ?/), Q(x, y) must therefore be a pair of solutions of Laplace's equation. Conversely, any solution of Laplace's equation may be taken for one of the functions P or Q, For example, let P (x, ?/) be a solution of that equation , the two equations (1), where Q is regarded as an unknown function, aie compatible, and the expression Uc.rt .. which is determined except for an arbitrary constant C, is an analytic function whose real part is P(x, y) It follows that the study of analytic functions of a complex van- able 3 amounts essentially to the study of a pan ot functions P(OJ, ?/), Q(a-, y) of two real variables x and ?/ that satisfy the lelations (1) It would be possible to develop the whole theoiy with- out making use of the symbol i * We shall continue, however, to employ the notation of Cauchy, but it should be noticed that there is no essential difference between the two methods Every theorem established for an analytic function /(#) can be expressed immediately as an equivalent theorem relat- ing to the pair of functions P and (2, and conversely Examples The function u = x 2 y 2 + 2xyi is an analytic function, for it satisfies the equations (1), and its denvative is 2x + 2yz = 2z , m fact, the func- tion is simply (x + yi) z = z 2 On the other hand, the expression v x yi is not an analytic function, for we have Av __ Ax i Ay _ """ "~ Az Ax + lAy . , Ay 1-f i Ax and it is obvious that the limit of the quotient Av/Az depends upon the limit of the quotient Ay /Ax If we put x == p cos w, y = p sin , and apply the formulae for the change of independent variables (I, 63, 2d cd , 38, 1st ed , Ex II), the relations (1) become /QA $P dQ <5Q dP (9) = p 1 = p > and the derivative takes the form dp * This is the point of view taken by the Geiman mathematicians who follow Eiemann I, 4] GENERAL PRINCIPLES 11 It is easily seen on applying these formulae that the function jpn = pm ( cos mtl} 4. i gin mw ) is an analytic function of z whose derivative is equal to s w i sin w) = 4. Functions analytic throughout a region. The preceding general statements are still somewhat vague, foi so far nothing has been said about the limits between which z may vary. A portion A of the plane is said to be connected, or to consist of a single piece, when it is possible to join any two points whatever of that portion by a continuous path which lies entirely in that portion of the plane. A connected portion situated entirely at a finite distance can be bounded by one or several closed curves, among which there is always one closed curve which forms the exterior boundary A portion of the plane extending to infinity may be composed of all the points exterior to one or moie closed curves ; it may also be limited by curves having infinite branches We shall employ the term region to denote a connected portion of the plane A function f(&) of the complex variable a? is said to be analytic * in a connected region A of the plane if it satisfies the following conditions 1) To every point & of A corresponds a definite value of /(#) , 2) /(#) is a continuous function of # when the point # varies in A, that is, when the absolute value of f(z + /0""/W approaches zero with the absolute value of h , 3) At every point & of A, f(z) has a uniquely determined deriva- tive /'() , that is, to every point corresponds a complex number / r () such that the absolute value of the difference n/ approaches zero when \h\ approaches zero Given any positive num- ber c, another positive number t\ can be found such that (4) i/^ + fc)-/^) if | h\ is less than ^ For the moment we shall not make any hypothesis as to the values of /(#) on the curves which limit A. When we say that a function is analytic in the interior of a region A bounded by a closed curve T * The adjective hotomorphic is also often used. TBAKS 12 ELEMENTS OF THE THEORY [I, 4 and on the boundary curve faelf, we shall mean by this that f(z) is analytic in a region Jl containing the boundary cmve T and the region A A function /() need not necessaiily be analytic throughout its region of existence It may have, in general, singular points, which may be of veiy varied types It would be out of place at this point to make a classification of these singular points, the veiy nature of which will appear as we proceed with the study of functions which we are now commencing 5. Rational functions. Since the rules which give the derivative of a sum, of a product, and of a quotient are logical consequences of the definition of a derivative, they apply also to functions of a complex variable The same is true of the vule for the derivative of a func- tion of a function Let u =sf(Z) be an analytic function of the complex variable Z , if we substitute for Z another analytic function < (#) of another complex varrable &, u is still an analytic function of the variable 2. We have, in fact, Aw _ A?t ^ z . A* ~~ &Z A* ' when |A&| approaches zero, |AJ| approaches zero, and each of the quotients AW/A-2T, A-^T/As approaches a definite limit Therefore the quotient A-w/A itself approaches a limit We have already seen (3) that the function is an analytic function of #, and that its derivative is mz m ~ l . This can be shown directly as in the case of real variables In fact, the binomial formula, which results simply from the properties of multi- plication, obviously can be extended in the same way to complex quantities Therefore we can write 1 1 A where m is a positive integer ; and from, this follows I, 6] GENERAL PRINCIPLES 13 It is cleat that the right-hand side has ma**- 1 for its limit when the absolute value of h appioaches zero. It follows that any polynomial with constant coefficients is an analytic function thioughout the whole plane A rational function (that is, the quotient of two polynomials P(, Q(z), which we may as weU suppose prime to each othei) is also in general an analytic function, but it has a certain number of singular points, the roots of the equation Q(z)= It is analytic in every region of the plane which does not include any of these points 6. Certain irrational functions. When a point & descubes a continu- ous curve, the coordinates x and y, as well as the absolute value p, vary in a continuous manner, and the same is also tiue of the angle, x FIG 30 36 provided the curve described does not pass through the origin. If the point & describes a closed curve, #, y, and p return to their original values, but for the angle o> this is not always the case. If the origin is outside the region inclosed by the closed curve (Eig. 3 &), it is evident that the angle will return to its original value , but this is no longer the case if the point & describes a curve such as M^NPM^ or MftpqM^ (Fig 3 ) In the first case the angle takes on its original value increased by 2 TT, and in the second case it takes on its original value increased by 4 TT. It is clear that # can be made to describe closed curves such that, if we follow the continuous variation of the angle along any one of them, the final value assumed by <D will differ from the initial value by 2 mr, where n is an arbitrary integer, posi- tive or negative. In general, when describes a closed curve, the 14 ELEMENTS OF THE TIIEOKY [I, 6 angle of 2 a returns to its initial value if the point a lies outside of the region bounded by that closed cuive, but the cuive described by & can always be chosen so that the final value assumed by the angle of & a will be equal to the initial value increased by 2n7r Let us now consider the equation (5) f = , where m is a positive integer To every value of 2, except 2 = 0, there are m distinct values of u which satisfy this equation and therefore correspond to the given value of z In fact, if we put & = p (cos <o + *> sin <o), u = r (cos < + I sm <), the relation (5) becomes equivalent to the following pan w<}> = o) + 27i TT From the first we have i = p l/wl , which means that > is the wth anth- metic root of the positive imnibei p , from the second we have To obtain all the distinct values of we Lave only to givo to the arbitrary integer 7c the m consecutive integral values 0, 1, 2, , m 1 , in this way we obtain expiessions for the w, roots of the equation (5) /^ ^f /<o + 27r\, . /W + 2&7TY1 (6) Wfc = ^[e OS (__^ , (* = 0,1, 2, ,w-l). It is usual to represent by 1M any one of these roots Wlien the variable z describes a continuous curve, each of these roots itself varies in a continuous manner If s describes a closed curve to which the oxigin is exteiior, the angle w comes back to its original value, and each of the roots U Q , u v , u m ^ describes a closed curve (Fig 4 a) But if the point z describes the curve JM Q NPM Q (Fig 3 5), GJ changes to <w + 2 TT, and the final value of the root u % is equal to the initial value of the root w, +1 Hence the arcs described by the different roots form a single closed curve (Fig 4 5) These m roots therefore undergo a cyclic permutation when the variable describes m the positive direction any closed curve with- out double points that incloses the origin. It is clear that by making * describe a suitable closed path, any one of the roots, starting from the initial value u^ for .example, can be made to take on for its final value the value of any of the other roots. If we wish to maintain continuity, we must then consider these m roots of the equation (5) I, 6] GENERAL PRINCIPLES 15 not as so many distinct functions of *, but a,s m distinct branches of the same function The point = 0, about which the permutation of the m values of u takes place, is called a critical point or a Iranc7i point 46 In order to consider the m values of u as distinct functions of &, it will be necessaiy to disiupt the continuity of these roots along a line proceeding from the origin to infinity. We can represent this bieak m the continuity very concretely as follows imagine that m the plane of 2, which we may legard as a thin sheet, a cut is made along a ray extending from the ongin to infinity, for example, along the ray OL (Fig. 5), and that then the two edges of the cut are slightly separated so that there is no path along which the variable & can move dnectly fiom one edge to the other Under these circum- stances no closed path whatever can inclose the origin, hence to each value of z corresponds a completely detexnuned value u % of the m roots, which we can obtain by tak- ing for the angle <> the value included between a and a 2 TT But it must be noticed that the values of u t at two points m, m f on opposite sides of the cut do not approach the same limit as the points approach the same point of the cut The limit of the value of u^ at the point m 1 is equal to the limit of the value of ^ 8 at the point m, multi- plied by [cos (2 vr/m) + 1 sin (2 7r/m)] Each of the roots of the equation (5) is an analytic function. Let u be one of the roots corresponding to a given value & Q , to a value of near # Q corresponds a value of u nea/i u Instead of trying to 16 ELEMENTS OF THE THEORY [I, 6 find the limit of the quotient (u u^/(z s ), we can determine the limit of its reciprocal z s u m u% and that limit is equal to mu*-\ We have, then, for the denvative m u m ~ l m z or, using negative exponents, , 1 -1 u 1 = 2 m w In order to be sure of having the value of the denvative which coi re- sponds to the root considered, it is bettei to make use of the expies- sion (l/w)(i05). In the interior of a closed curve not containing the origin each of the determinations of V& is an analytic function The equation u m = A (2 a) has also m roots, which permute themselves cyclically about the critical point 2 = a Let us consider now the equation (7) u* = A(z- e,)(* - 2 ) - (* - O, where e v e# , e n are n distinct quantities We shall denote by the same letters the points which represent these n quantities Let us set A = R (cos a + t sin a), e k = p k (cos <o L + i sin w^.), (>fe = 1, 2, , TI), ^ = r(cos + i sin 0), where o) ft represents the angle which the straight-line segment e k z makes with the direction Ox. From the equation (7) it follows that Pn , = -a) 1 . <^ hence this equation has two roots that are the negatives of each other, (8) N*P />)*[ + >i + + u. + 2 cos - " I, 7] GENERAL PRINCIPLES 17 When the variable z describes a closed curve C containing within it p of the points e v e^ , e n , p of the angles <a v <*># , o n will increase by 27r, the angle of u^ and that of w 2 will therefoie in- crease by 2 J7r If P 1S even, the two loots leturn to their initial values , but if p is odd, they aie peimuted In particular, if the curve incloses a single point e t , the two loots will be permuted. The n points e t are branch points In order that the two roots x and u 2 shall be functions of z that are always uniquely determined, it will suffice to make a system of cuts such that any closed curve whatever will always contain an even number of critical points. We might, for example, make cuts along lays proceeding from each of the points e % to infinity and not cutting each other But there are many other possible arrangements If, for example, there are four criti- cal points e v e 2 , 8 , e^ a cut could be made along the segment of a straight line e^ and a second along the segment e^e. 7. Single-valued and multiple-valued functions. The simple exam- ples which we have ]ust treated bring to light a very important fact. The value of a function f(z) of the variable & does not always depend entirely upon the value of z alone, but it may also depend in a cer- tain measure upon the succession of values assumed by the variable z in passing from the initial value to the actual value in question, or, in other words, upon the path followed by the variable z. Let us return, for example, to the function u = V#. If we pass from the point Jf to the point M by the two paths MJKM and M Q PM (Fig. 3 #), starting in each case with the same initial value for u, we shall not obtain at M the same value for u, for the two values obtained for the angle of z will differ by 2 TT. We are thus led to introduce a new distinction. An analytic f unction f(z) is said to be singles alued* in a legion A when all the paths in A which go from a point z^ to any other point whatever z lead to the same final value for /(#). When, however, the final value of f(z) is not the same for all possible paths in -4, the function is said to be 'multiple-valued t A function that is analytic at every point of a region A is necessarily single-valued in that region. In general, in order that a function f(z) be single- valued in a given region, it is necessary and sufficient that the func- tion return to its original value when the variable makes a circuit of * In French the term umforme or the term rnonodrome is used TRANS t In French the term multiforme is used TEANS 18 ELEMENTS OF THE THEORY [l,7 any closed path whatevei If, in fact, m going fiom the point A to the point B by the two paths AMB (Fig 6) and ANB, we airive in the two cases at the point B with the same determination of f(z), it is obvious that, when the vanable is made to describe the closed cuive A MBNA, we shall return to the point A with the initial value of /() Conversely, let us suppose that, the vanar ble having descubed the path AMBNA, we return to the point of departure with the initial value ^ ; and let it^ be the value of the function at the point B after z has described the path AMB. When z deseiibes the path BNA, the function starts with the value u t and amves at the value ^ ; then, conversely, the path ANB will lead from the value U Q to the value u^ that is, to the same value as the path AMB It should be noticed that a function which is not single-valued in a legion may yet have no critical points in that region Consider, for example, the poition of the plane included between two concentric cir- cles (\ r" having the origin for center. The function u = r- v in has no cutical point in that region , still it is not single-valued in that region, for if z is made to describe a concentric circle between C and 6 y ', the function g 1/m will be multiplied by cos (2 TT/WI) -f i sin (2 ir/m) JL POWER SERIES WITH COMPLEX TERMS ELEMENTARY TRANSCENDENTAL FUNCTIONS 8. Circle of convergence. The reasoning employed in the study of power series (Vol I, Chap IX) will apply to power series with complex terms ; we have only to replace in the reasoning the phrase " absolute value of a real quantity" by the corresponding one, " absolute value of a complex quantity " We shall recall briefly the theorems and results stated there Let (9) + V + a^ + + <v~ n + be a power series in which the coefficients and the variable may have any imaginary values whatevei Let us also consider the series of absolute values. (10) AQ+A^+A^-I \~A n i+ , where ^ = [aj, r- = |#] We can prove (I, 181, 2d ed ; 177, 1st ed ) the existence of a positive number 12 such that the series I, 8] POWER SERIES WITH COMPLEX TERMS 19 (10) is convergent foi eveiy value of )><R, and diveigent for every value of r>R The number R is equal to the leciprocai of the greatest limit of the terms of the sequence and, as particular cases, it may be zero or infinite Prom these pioperties of the number R it follows at once that the series (9) is absolutely convergent when the absolute value of z is less than R It cannot be convergent for a value Z Q of & whose abso- lute value is gieater than R, for the series of absolute values (10) would then be convergent for values of r greater than R (I, 181, 2d ed , 177, 1st ed ) If, with the origin as center, we descube in the plane of the variable z a circle C of ladius R (Fig. 7), the power seiies (9) is absolutely convergent for every value of z inside the circle (7, and divergent for every value of z outside , for this reason the circle is called the circle of convergence In a point of the circle itself the series may be convergent or divergent, according to the particular series * In the intenoi of a circle <?' concentric with the first, and with a ladius R 1 less than 72, the series (9) is uniformly convergent For at every point within C 1 we have evidently and it is possible to choose the integer n so large that the second member will be less than any given positive number e, whatever %> may be From this we conclude that the sum of the series (9) is a continuous function /() of the variable % at every point within the circle of convergence ( 2) By differentiating the seiies (9) repeatedly, we obtain an unlimited number of power series, /,(*), /,(*), . , /;(*), . , which have the same circle of convergence as the first (I, 183, 2d ed. ; 179, 1st ed ) We prove in the same way as in 184, 2d ed , that f^z) is the denvative of /(), and in general that f n (z) is the derivative * Let/(z) = SanZ* 1 be a power series whose radius of convergence JR is equal to 1 If the coefficients a , a^ a s , , are positive decreasing numbeis such that a n ap- proaches zeio when n inci eases indefinitely, the series is convergent in every point of the circle of convergence, except perhaps for z= 1 In fact, the seiies Sz, where 1 2 1 = 1, is indeterminate except for zi, foi the absolute value of the sum of the fiist n terms is less than 2/| 1 - z | , it will suffice, then, to apply the reasoning of 166, Vol I, based on the generalized lemma of Abel In the same way the sei les a - % z + a 2 z 2 - which is obtained fiom the piecedmg by replacing z by - z, is convergent at all the points of the circle 1*1-1, except perhaps f or z = - 1 (Cf I, 166 ) 20 ELEMENTS OF THE THEORY of f n _i(z) JSvery power series represents therefore an analytic func- tion in the interior of its circle of convenience There is an infinite sequence of denvatives of the given function, and all of them aie analytic functions in the same ciicle Given a point # inside the circle C, let us draw a circle c tangent to the circle C in the interior, with the given point as cen- ter, and then let us take a point a + A inside c , if r and p are the absolute values of and h, we have r + p<R (Fig. 7). The sum /( 4- A) of the series is equal to the sum of the double series FIG. 7 + a n (n 1) 1 2 when we sum by columns But this series is absolutely convergent, for if we replace each term by its absolute value, we shall have a double series of positive terms whose sum is A i + A 1 (r + p) + .+ A n (r + ,) + .... We can therefore sum the double series (11) by rows, and we have then, for every point # + h inside the circle c } the relation (12) The series of the second member is surely convergent so long as the absolute value of h is less than R r, but it may be convergent in a larger circle Since the functions / a (), / 2 ()j > /()? are equal to the successive derivatives of /(*), the formula (12) is identical with the Taylor development. If the series (9) is convergent at a point Z of the circle of con- vergence, the sum/() of the series is the limit approached by the sum /(#) when the point approaches the point Z along a radius I, 9] POWER SERIES WITH COMPLEX TERMS 21 which, terminates in that point. We prove this just as in Volume I ( 182, 2d ed , 178, 1st ed ), by putting * == 6Z and letting in- crease from to 1 The theorem is still true when z, remaining inside the circle, approaches Z along a curve which is not tangent at Z to the cncle of convergence * When the radius R is infinite, the circle of convergence includes the whole plane, and the function f(&) is analytic for every value of z We say that this is an integral function , the study of tran- scendental functions of this kind is one of the most important objects of Analysis t We shall study 111 the following paragraphs the classic elementary transcendental functions 9. Double series Given a power series (9) with any coefficients whatever, we shall say again that a second power series ar B z re . whose coefficients are all real and positive, dominates the first series if f 01 every value of n we have | a n \ =i cc n . All the consequences deduced by means of dominant functions (I, 186-189, 2d ed , 181-184, 1st ed ) follow without modification in the case of complex variables We shall now give another application of this theory Let (13) / () + f l (z) +/,() + + f n (z) + . . be a series of which each term is itself the sum of a power series that converges in a cncle of radius equal to or gi eater than the number E > 0, Suppose each term of the series (13) replaced by its development according to powers of z , we obtain thus a double series in which each column is formed by the development of a f unction f t (z) When that series is absolutely convergent for a value of z of absolute value />, that is, when the double series is convergent, we can sum the first double series by rows for every value of z whose absolute value does not exceed p. We obtain thus the development of the sum F(z) of the series (13) in powers of z, - + a m +..., (n = 0, 1, 2, ) This proof is essentially the same as that for the development of f(z -f h) in powers of h Suppose, for example, that the series f t (z) has a dominant function of the form M t r/(r z), and that the series SJf z is itself convergent In the double * See PICABD, Traite d'Analyse, Vol. II, p 73 t The class of integral functions includes polynomials as a special case If there are an infinite number of terms in the development, we shall use the expression integral transcendental function TRANS 22 ELEMENTS OP THE THEORY [i, 9 series the absolute value of the general term is less than M t \z\ n /i" If | z | < r, the series is absolutely convergent, foi the senes of the absolute values is convergent and its sum is less than r'ZM l /(r | z |) 10. Development of an infinite product m power series. Let be an infinite product wheie each of the functions u % is a continuous function of the complex variable z in the region D If the series SlT t , where Z7 t = | u, |, is uniformly conveigent in the region, F(z) is equal to the sum of a senes that is uniformly convergent in Z), and therefoie repiesents a continuous function (I, 175, 176, 2d ed.). When the functions u l aie analytic functions of 2, it fol- lows, fiom a general theorem which will be demonstrated latei ( 39), that the same is true of F(z) JTor example, the infinite product represents a function of z analytic thioughout the entire plane, for the series S|s| 2 /w 2 is uniformly convergent within any closed curve whatevei This product is zero for z = 0, 1, 2, and for these values only. We can prove directly that the product F(z) can be developed m a power senes when each of the functions u % can be developed m a power senes such that the double senes is convergent for a suitably chosen positive value of r. Let us set, as in Volume I ( 174, 2d ed ), It is sufficient to show that the sum of the series (14) which is equal to the infinite product F(z), can be developed in a power series. Now, if we set < it is clear that the product is a dominant function for v n It is therefore possible to arrange the series (14) according to powers of z if the following auxiliary senes (15) J + i + +<+- can be so arranged If we develop each term of this last series in power series, we obtain a double series with positive coefficients, and it is sufficient for our purpose to I, 11] POWER SERIES WITH COMPLEX TERMS 23 prove that the double series converges when z is replaced by r. Indicating by UK and V' n the values of the functions u n and i for z = r, we have and therefore or, again, When n increases indefinitely, the sum UQ + - + U^ approaches a limit, since the series SlT^ is supposed to be convergent The double senes (15) is then absolutely convergent if | z \ s r , the double senes obtained by the development of each term v n of the series (14) is then a f ortioii absolutely convergent within the circle C of radius r, and we can arrange it according to integral powers of z The coefficient b p of zP m the development of F(z) is equal, from the above, to the limit, as n becomes infinite, of the coefficient 6pof z* in the sum i? + v x + -f v n , or, what amounts to the same thing, in the development of the product Hence this coefficient can be obtained by applying to infinite products the ordinary rule which gives the coefficient of a power of z in the product of a finite number of polynomials For example, the infinite product can be developed according to powers of z if | z \ < 1 Any power of z whatever, say Z N , will appear in the development with the coefficient unity, f 01 any posi- tive integer N can be written in one and only one way in the form of a sum of powers of 2 We have, then, if | z \ < I , (16) 1 z which can also be very easily obtained by means of the identity 11. The exponential function. The arithmetic definition of the ex- ponential function evidently has no meaning when the exponent is a complex number. In order to generalize the definition, it will be necessary to start with some property which is adapted to an exten- sion to the case of the complex variable We shall start with the property expressed by the functional relation Let us consider the question of determining a power series /(), con- vergent in a circle of radius R, such that (17) /( + *')=/</(*') when, the absolute values of , ', a + s' are less than R, which will 24 ELEMENTS OF THE THEORY [I, 11 surely be the case if \\ and |'| are less than R/2 If we put 2' = in the above equation, it becomes Hence we must have /(O) = 1, and we shall write the desired series Let us replace successively in that series & by X, then by X', where X and X' are two constants and t an auxiliary variable ; and let us then multiply the resulting series This gives On the other hand, we have The equality /(Xi + X r O=/(XO/(A. f #) is to hold for all values of X, X', t such that |X| < 1, |X'| < 1, |*| < /2/2. The two series must then be identical, that is, we must have and from this we can deduce the equations a n = a n _ l a 1 , a w =a n 2 a 2 , , all of which can be expressed in the single condition where jp and gf are any two positive integers whatever. In order to find the general solution, let us suppose q = 1, and let us put successively p = 1, p = 2, JK> = 3, ; from this we find 2 = af , then <& 8 = 0,^= aj, , and finally a n = aj. The expressions thus obtained satisfy the condition (18), and the series sought is of the form I, 11] POWER SERIES WITH COMPLEX TERMS 25 This series is convergent in the whole plane, and the relation /(* + *')=/(*) A* 1 ) is true for all values of & and ^ The above series depends upon an arbitrary constant a v Taking a l = 1, we shall set so that the geneial solution of the given problem is e a i z The inte- gral function e z coincides with the exponential function e x studied in algebra when is real, and it always satisfies the relation whatever z and z* may be The derivative of e z is equal to the func- tion itself. Since we may write by the addition formula in order to calculate e z when & has an imaginary value x + yi, it is sufficient to know how to calculate e vt Now the development of e m can be written, grouping together terms of the same kind, We recognize in the second member the developments of cos y and of siny, and consequently, if y is real, Replacing e^ by this expression in the preceding formula, we have (19) &* + Vl = e*(cos y + i sin y) , the function e*+w has e* for its absolute value and y for its angle. This formula makes evident an important property of e* , if z changes to & + 2 m, x is not changed while y is increased by 2 TT ? but these changes do not alter the value of the second member of the formula (19) We have, then, that is, the exponential function e* has the period 2 m Let us consider now the solution of the equation e* = A, where A is any complex quantity whatever different from zero Let p and <u be the absolute value and the angle of A , we have, then, ga+y* = a x (cos y + ^ sm y) = p (cos o> + i sin <o), 26 ELEMENTS OF THE THEORY [I, ll from which it follows that e x = p, y = <o + 2 JCTT. From the first relation we find x = log p, where the abbreviation log shall always be used foi the natmal logarithm of a leal positive number. On the other hand, y is determined except for a multiple of 2 TT If A is zero, the equation & = leads to an impossibility Hence the equation e* = A, wJiere A is not zero, has an ^nfin^te num- ber of roots given ly the expression logp 4- i( + 27c7r), the equation Q Z = has no roots, real or imaginary Note We might also define e z as the limit approached by the poly- nomial (1 + &/m) m when m becomes infinite The method used in algebia to prove that the limit of this polynomial is the series e* can be used even when & is complex 12. Trigonometric functions. In order to define sin & and cos # when 2 is complex, we shall extend directly to complex values the series established for these functions when the variable is real Thus we shall have (20) These are integral transcendental functions which have all the properties of the trigonometric functions. Thus we see from the formulae (20) that the derivative of sin # is cos #, that the derivative of cos & is sin #, and that sm % becomes sin , while cos % does not change at all when is changed to & These new transcendental functions can be brought into very close relation with the exponential function In fact, if we write the ex- pansion, of e* 1 , collecting separately the terms with and without the factor L 21 ' 4! ' we find that that equality can be written, by (20), in the form &*> ss cos * + i sm . Changing to 2, we have again e~~** = cos r and from these two relations we derive I, 12] POWER SERIES WITH COMPLEX TERMS 27 cos s = - - - > sins 2 ' OA "~ 2i These are the well-known formulae of Euler which express the trigonometric functions in terms of the exponential function They show plainly the periodicity of these functions, for the right-hand sides do not change when we replace z by z 4- 2 TT Squaring and adding them, we have cos 2 z 4- sin 2 2 = 1 Let us take again the addition formula e (z+ * 0t = e 2 *e a/l , or cos (z + z') 4- i sin (z 4- z') = (cos z 4- i sin z) (cos z' 4- i sin z 1 ) = cos 2 cos z 1 sin sin z' i-^ (sin 2 cos 2' + sin z 1 cos 2), and let us change z to 2, 2' to 2'. It then becomes cos (2 4- 2 1 ) * sin (2 4- 2*) = cos z cos 2' sin z sin ' t(sin 2 cos # f 4- sin z' cos 2), and from these two formulae we derive cos (z 4- #') =5 cos 2 cos z f sin 2 sin z' sin (2 4- 2') = sin z cos #' 4- sin z cos 2'. The addition formulae and therefore all their consequences apply for complex values of the independent variables Let us determine, for example, the real part and the coefficient of i in cos (x + yi) and sin (x + 2/1). We have first, by Euler's formulae, . e e , cos 7^ = - 5 - = coshy, 5 - , = whence, by the addition formulae, cos (x + yi) = cos a; cos y sin oc sin yt = cos x cosh y t sin x sinh y, sin (x + y&) = sin x cos iy + os x sin 2/ = sin x cosh y + 1 cos x sinh T/ The other trigonometric functions can be expressed by means of the preceding For example, sin 2 10 st e~ s * tan 2 = - = T -TT-: - z ' cos 2 * 6**+ e"* 1 which may be written in the form The right-hand side is a rational function of e 2 **; the penod of the tangent is therefore TT. 28 ELEMENTS OF THE THEORY [I, 13 13. Logarithms. Given, a complex quantity #, different fiom zero, we have already seen ( 11) that the equation e u = has an infinite numher of roots Let u = x + yi, and let p and u> denote the absolute value and angle of &, respectively Then we must have &* = p } f/ = 0) + 2 A 7T Any one of these roots is called the logarithm of and will be denoted by Log () We can write, then, Log (s) = log p + i (o> + 2 //TT), the symbol log being leserved for the ordinal y natuial, or Napierian, logarithm of a real positive number Every quantity, real or complex, different fiom zero, has an infinite number of logarithms, which form an anthmetic progies- sion whose consecutive terms diffei by 2 m In particular, if is a real positive number x, we have <o = Taking k = 0, we find again the ordinary logarithm , but there aie also an infinite number of complex values for the logarithm, of the form logos 4- 2 kiri If z is real and negative, we can take <o = TT , hence all the determinations of the logarithm are imaginary Let &' he another imaginary quantity with the absolute value />' and the angle o>'. We have Log (>') = log p' + 4 (' + 2 /C'TT) Adding the two logarithms, we obtain Log(s)+ Log(>')= logpp f 4- *|> + ' + 2(A + /C')TT] Since pp* is equal to the absolute value of ##', and o> + CD' is equal to its angle, this formula can be written in the form Log (*) + Log (V) = Log <X), which shows that, when we add any one whatever of the values of Log(#) to any one whatever of the values of Log(V), the sum Is one of the determinations of Log(^ f ). Let us suppose now that the variable s describes in its plane any continuous curve whatever not passing through the origin; along this curve p and <o vary continuously, and the same thing is true of the different determinations of the logarithm But two quite distinct cases may present themselves when the variable & traces a closed curve When starts from a point # and returns to that point after having described a closed curve not containing the origin within it, the angle o> of takes on again its original value <D O , and the different 1, 13] POWER SERIES WITH COMPLEX TERMS 29 determinations of the loganthm eoine back to their initial values. If we represent each value of the logarithm by a point, each of these points traces out a closed cuive On the contrary, if the vanable z describes a closed curve such as the curve M^NMP (Fig. 3 #), the angle increases by 27r, and each determination of the logarithm returns to its initial value increased by 2iri In general, when & describes any closed curve whatever, the final value of the logarithm is equal to its initial value increased by 2k7ri, wheie k denotes a positive or negative integer which gives the number of i evolutions and the direction through which the radius vector joining the origin to the point z has tinned. It is, then, impossible to considei the dif- ferent determinations of Log (z) as so many distinct functions of z if we do not place any restriction on the vanation of that vanable, since we can pass continuously from one to the other They aie so many branches of the same function, which aie permuted among themselves about the critical point z = In the interior of a region which is bounded by a single closed curve and which does not contain the origin, each of the determinations of Log (2) is a continuous single-valued function of z. To show that it is an analytic function it is sufficient to show that it possesses a unique derivative at each point Let z and z^ be two neighboring values of the variable, and Log(#), Log (24) the corresponding values of the chosen deteimmation of the logarithm When & l approaches 2, the absolute value of Log ( a ) Log (z) approaches zero Let us put Log (*) = w, Log (^ = u^ , then When u^ approaches u, the quotient e tt i - e u u^ u approaches as its limit the derivative of e w , that is, e u or z. Hence the logarithm has a uniquely determined derivative at each point, and that derivative is equal to 1/2. In general, Log (2 a) has an infinite number of determinations which permute themselves about the critical point z = a, and its derivative is l/(s a) The function z m , where m is any number whatever, real or complex, is defined by means of the equality 30 ELEMENTS OF THE THEORY [I, 13 Unless m be a real lational number, this function possesses, just as does the logarithm, an infinite number of deteiminations, which per- mute themselves when the variable turns about the point & = It is sufficient to make an infinite cut along a ray fiom the origin in order to make each branch an analytic function in the whole plane The derivative is given by the expression and it is clear that we ought to take the same value for the angle of z in the function and m its derivative 14. Inverse functions : arc sin z, arc tan z. The inverse functions of sin s, cos 2, tan & are defined in a similar way. Thus, the function u = arc sin & is defined by the equation 2 = sin u In order to solve this equation for u, we write 6 &~ m _ 2tu 1 *~ 2^ ~T^' and we are led to an equation of the second degree, (22) 17 2 - 2^-1=0, to deteimine the auxiliary unknown quantity U = e ui . We obtain from this equation (23) U = fa Vl s a , or i (24) u = arc sm 2 = - Log (i& Vl # 3 ) The equation z = sin ^ has therefore two sequences of roots, which arise, on the one hand, from the two values of the ladical Vl # 2 , and, on the other hand, from the infinite number of determinations of the logarithm But if one of these determinations is known, all the others can easily be determined from it Let U 1 = p'e 11 *' and U" = p"e lw// be the two roots of the equation (22) ; between these two roots exists the relation 17 f i7"= 1, and therefore p'p" = 1, o)'-h o>" = (2 w + 1) TT. It is clear that we may suppose <o" = TT o> f , and we have then Log (U 1 ) = log P ' + <(' + Log (17") = - log P ' + j (w - o>' + 2 t"ir). I, 14] POWER SERIES WITH COMPLEX TERMS 31 Hence all the determinations of arc sin 2 are given by the two formulae axe sins = CD' + 2 &'TT i log//, arc sing = TT + 2 &"TT <o' + and we may write (A) arc sin = u f + (B) arc sin* = (2 where %' = a/ ^ logp'. When the variable describes a continuous curve, the various determinations of the logarithm in the formula (24) vary in general in a continuous manner. The only critical points that are possible are the points 2 = 1, around which the two values of the radical Vl g 2 are permuted, there cannot be a value of & that causes i Vl to vanish, for, if there were, on squaring the two sides of the equation iz = Vl # 2 we should obtain 1=0 Let us suppose that two cuts are made along the axis of reals, one going from oo to the point 1, the other from the point + 1 to + oo . If the path described by the variable is not allowed to cross these cuts, the different determinations of arc sins aie single-valued functions of 2. In fact, when the variable describes a closed curve not crossing any of these cuts, the two roots ?7 ; , U" of equation (22) also describe closed curves. None of these curves contains the origin in its interior. If, for example, the curve described by the root U 1 contained the origin in its interior, it would cut the axis Oy in a point above Ox at least once Corresponding to a value of U of the form ia(a > 0), the relation (22) determines a value (1 + cf)/2 a for s, and this value is real and > 1 The curve described by the point s would therefore have to cross the cut which goes from + 1 to + oo The different determinations of arc sin z are, moreover, analytic functions of z * Eor let u and u be two neighboring values of * If we choose in C r =i2+Vl~2 2 the determination of the radical which reduces to 1 when 2=0, the real part of U remains positive when the variable z does not cross the cuts, and we can put U= .Re**, where $ lies between --7T/2 and +<rr/2 The cor- responding value of (1/i) Log CT, namely, 7 is sometimes called the principal value of arc sm z It reduces to the ordinary deter- mination when z is real and hes between - 1 and + 1. 32 ELEMENTS OF THE THEORY [I, i* arc sing, corresponding to two neighboring values z and ^ of the variable We have Uj U U l U #j # sin u^ sin u When the absolute value of it l u approaches zero, the preceding quotient has for its limit 1 1 cos u The two values of the derivative correspond to the two sequences of values (A) and (B) of arc sin & If we do not impose any lestnction on the variation of #, we can pass fiom a given initial value of arc sins to any one of the deter- minations whatever, by causing the variable & to describe a suitable closed curve In fact, we see fiist that when # descubes about the point = 1 a closed curve to which the point z = 1 is exterior, the two values of the radical Vl "* are pei muted and so we pass fiom a determination of the sequence (A) to one of the sequence (B) Suppose next that we cause z to descube a circle of radius R (R > 1) about the origin as center, then each of the two points U\ U" describes a closed curve. To the point z= + R the equation (22) assigns two values of U, U' = m^ U" = ^|3, where a and ft are positive , to the point ^ = J2 there coriespond by means of the same equation the values U 1 = to;', U' 1 = ip 1 , where a' and {? are again, positive Hence the closed curves described by these two points U', T7" cut the axis Oy in two points, one above and the other below the point , each of the logarithms Log (7"'), Log(?7") increases or diminishes by 2?r4 In the same way the function aic tan a is defined by means of the relation tan u = 2, or 1 e* - 1 whence we have 2ta __ 3 + iz 6 - and consequently J 1 J arc tan z = ^ This expression shows the two logarithmic critical points i of the function arc tan &. When the variable z passes around one of these points, Log [(i z)/(i + #)] increases or diminishes by 2 iri, and arc tan increases or diminishes by TT I, 15] POWER SERIES WITH COMPLEX TERMS 33 15. Application to the integral calculus. The derivatives of the func- tions which we have just defined have the same form as when the variable is real Conveisely, the rules for finding primitive functions apply also to the elementaiy functions of complex variables Thus, denoting by ff(z)dz a function of the complex variable & whose derivative is /(s), we have Adz = _ A 1 (2 a) m m 1 (2 a)" 1 " 1 ^ =A Log(s a). These two formulae enable us to find a primitive function of any rational function whatever, with real or imaginary coefficients, pro- vided the roots of the denominator aie known. Consider as a special case a rational function of the real variable x with real coefficients If the denominator has imaginary roots, they occur in conjugate pairs, and each root has the same multiplicity as its conjugate. Let a + fti and a fii, be two conjugate roots of multiplicity p In the decomposition into simple fractions, if we proceed with the imaginary roots just as with the real roots, the root a + fii will furnish a sum of simple fractions M l + N l i M 2 + N 2 i M p + N p l '' x-a-fti (x-a-pi)*' (x-a- fii)* and the root a fti will furnish a similar sum, but with numerators that are conjugates of the former ones. Combining in the primitive function the terms which come from the corresponding fractions, we shall have, if p >1, p - 1 (* - - fa)"- 1 (x-a and the numerator is evidently the sum of two conjugate imaginary polynomials. If p = 1, we have CM^ + N^ J x-a-fii L + JVjt) Log [(as -a) - #.] + (M t - Nj,) Log [(* - a) + i] 34 ELEMENTS OF THE THEORY [I, 15 If we replace the logarithms by their developed expressions, there remains on the right-hand side M l log [(a? - of + /3 2 ] + 2 ^ aic tan ^f-^ It suffices to replace B , TT , x a arc tan c by 77 arc tan - x a J 2 p in order to express the result in the form in which it is obtained when imaginary symbols are not used Again, consider the indefinite integral dx which has two essentially different forms, according to the sign of A The introduction of complex variables reduces the two forms to a single one In fact, if in the formula we change x to ^x, there results dx 1. and the right-hand side represents precisely arc sin a?. The introduction of imaginary symbols in the integral calculus enables us, then, to reduce one formula to another even when the relationship between them might not be at all apparent if we were to remain always in the domain of real numbers We shall give another example of the simplification which comes from the use of imagmaries If a and 1) are real, we have J X ~~ a + bi~ < Equating the real parts and the coefficients of i, we have at one stroke two integrals already calculated (I, 109, 2d ed. ; 119, 1st ed.): _ , tf* (a cos bx + b sin bx) e cos bx dx = ^ 5 -: L j a a + lr 7 7 ^^sinftaj & cos foe) e sin bx dx = ii $ , / - I, 16] POWER SERIES WITH COMPLEX TERMS 85 In the same way we may reduce the integrals / x m e ax cog fa tf Xy I xm e ax g^ fa fa to the integral fx m e <a+bl "> x dx, which can be calculated by a succession of integrations by parts, where m is any integer. 16 Decomposition of a rational function of smz and cos^ into simple elements. Given a rational function of sin z and cos 2, .F(siii, cos #), if in it we replace sins and cos 2 by their expiessions given by Euler's formula, it becomes a rational function R() of t = e zl This function R (), decomposed into simple elements, will be made up of an integral part and a sum of fractions coming from the roots of the denominator of R (t) If that denominator has the root t = 0, we shall combine with the integral part the fractions aris- ing from that root, which will give a polynomial or a rational function R l (j)~'SK m t' in , where the exponent m may have negative values. Let t = a be a root of the denominator different from zero. That root will give rise to a sum of simple fractions The root a not being zero, let a: be a root of the equation e* 1 = a, , then !/( a) can be expressed very simply by means of ctn [(# #) We have, in fact, . * or .< eta. ^ whence it follows that 1 1 Hence the rational fraction /(#) changes to a" polynomial of degree w in ctn [(s ) The successive powers of the cotangent up to the nth can be ex- pressed in turn in terms of its successive derivatives up to the (n l)thj we have first d ctn z dz sin 2 * l ctn 2 *, 36 ELEMENTS OF THE THEORY [I, 16 winch enables us to express ctn 2 s 111 terms of d(cknz)/dz, and it is easy to show, by mathematical induction, that if the law is ti ue up to ctn w #, it will also be true for ctn tt+1 # The preceding polynomial of degree n in ctn[( &)/2] will change to a linear expression in ctn[(s #)/2] and its derivatives, or d n ~ l Let us proceed in the same way with all the roots b, c, , / of the denominator of R () different from zero, and let us add the results obtained after having replaced t by e zl in 11$) The given rational function F(sms, cos 2) will be composed of two parts, (25) .F( sin a, cos a) = $ () + * () The function $(#), which corresponds to the integral part of a rational function of the vanable, is of the form (26) $ (*) = C + S (tf /tt cos MS + j8 OT sin mst) 9 where 7?i is an integer not zero On the other hand, ^(), which cor- responds to the fractional part of a rational function, is an expression of the form It is the function ctn[( a)/2] which here plays the role of the simple element, just as the fraction !/( a) does for a rational function The result of this decomposition of /''(sin , cos ) is easily integrated , we have, in fact, (27) and the other terms are integrable at once In order that the primi- tive function may be periodic, it is necessary and sufficient that all the coefficients C, j4 v $ v be zero. In practice it is not always necessary to go through all these suc- cessive transformations in order to put the function j?(sin #, cos ) into its final form (25) Let # be a value of * which makes the function F infinite We can always calculate, by a simple division, the I, 16] POWER SERIES WITH COMPLEX TERMS 37 coefficients of (z a)" 1 , (z a)~ 2 , , in the part that is infinite for z = a (I, 188, 2d ed , 183, 1st ed ) On the other hand, we have where P (z #) is a power series , equating the coefficients of the successive poweis of (z a)- 1 in the two sides of the equation (25), we shall then obtain easily jj^ X, - -, Jl n Consider, for example, the function l/(cos # cos a) Setting e zl = t, e al = a, it takes the foim 2 at The denominator has the two simple loots t = a, * = I/a, and the numerator is of lower degree than the denominator. We shall have, then, a decomposition of the form cos z cos a In order to determine jl, let us multiply the two sides by z a;, and let us then put z = a. This gives 1 /T = l/(2 sin a). In a similar manner, we find $ = V(2 sin <*) Replacing ^ and $ by these values and setting z = 0, it is seen that C = 0, and the formula takes the form 1 = 1 cos z cos a ~" 2 sin a Let us now apply the general method to the integral powers of sin z and of cos 2 We have, for example, Combining the terms at equal distances from the extremities of the expansion of the numerator, and then applying Euler's f ormulse, we find at once (2 cosz) = 2 cosmz + 2m cos(m 2)2 + 2 m \ "" ^cos(m 4)z + 1 2 If m is odd, the last term contains cos z , if m is even, the term which ends the expansion is independent of z and is equal to m '/ [(m/2) '] 2 . In the same way, if m is odd, (2 1 smz)* = 2 * sin mz 2 ww- sm(m 2)2 + 2z m ^" ^sin (m 4)2 - - - , and if m is even, m - ' ; 2mcos(m 2)2+ + ( I) 2 These formulae show at once that the primitive functions of (sm2) m and of (cos z) m are periodic functions of z when m is odd, and only then 38 ELEMENTS OF THE THEORY [I, 16 Note When the function F(8inz, eos#) has the penod IT, we can express it rationally in terms of &*** and can take foi the simple elements ctn (0 a), ctii(# /3), * 17. Expansion of Log (1 + *). The transcendental functions which we have defined are of two kinds . those which, like e*j sin , cos *, are analytic in the whole plane, and those which, like Log &, arc tan ,, have singular points and cannot be represented by developments in power series convergent m the whole plane Nevertheless, such functions may have developments holding for certain paits of the plane We shall now show this for the logarithmic function Simple division leads to the elementary formula and if |*| <1, the remainder z*+ l /(l+z) approaches zero when n increases indefinitely Hence, m the interior of a circle C of radius 1 we have -i Let F(&) be the senes obtained by integrating this series term by term -, ^2 ^8 *4 .tt+l ^)-I-5 + F-I-- +<- 1 )%7+T + " ; this new series is convergent inside the unit circle and represents an analytic function whose derivative F'(z) is 1/(1 + z) We know, however, a function which has the same derivative, Log (1 + 0) It follows that the difference Log (1 -f 0) F() reduces to a constant.* In order to determine this constant it will be necessary to fix pre- cisely the determination chosen for the logarithm. If we take the one which becomes zero for = 0, we have for every point inside (28) Log(l + ) = |-| + |'-| + .... Let us join the point A to the point M, which represents (Fig. 8) The absolute value of 1 + * is represented by the length r = AM. Por the angle of 1 + z we can take the angle a which AM makes with AO, an angle which lies between 7r/2 and + Tr/2 as long as the point M remains inside the circle (7. That determination of the * In order that the denvative of an analytic function JT+ Yi "be zero, it is neces- sary that we haye (3) 0J5r/&e0, SF/Saj0, and consequently Xand Tare therefore constants. I, 17] POWER SERIES WITH COMPLEX TERMS 39 logarithm which becomes zero for & = is log r -f- ice ; hence the formula (28) is not ambiguous FIG 8 Changing 2 to s in this formula and then subtracting the two expressions, we obtain If we now replace z by iz } we shall obtain again the development of arc tan # The series (28) remains convergent at every point on the circle of convergence except the point A (footnote, p. 19), and consequently the two series cos20 , cos 30 cos40 f sm 20 sm 3 sm 4 ._ (._ _+ ... are both convergent except f or = (2 k + 1) ir (cf I, 166) By Abel's theorem the sum of the series at M' is the limit approached by the sum of the series at a point M as M approaches M' along the radius OJkP If we suppose always between TT and + TT, the angle a will have for its limit 0/2, and the absolute value AM will have for its limit 2 cos (0/2) We can therefore write / ft 0\ /, cos20 cos30 cos40 , log (2cos-)=cos0 _ + _+ , \ ] ft O * If In the last formula we replace by B IT, we obtain again a formula pre- viously established (I, 204, 2d ed , 198, 1st ed.) 40 ELEMENTS OF THE THEORY [I, is 18. Extension of the binomial formula. In a fundamental paper on power series, Abel set for himself the problem of determining the sum of the convergent series H , m , m ( m = m(m~l) 2 i '* + for all the values of m and #, real or imaginary, provided we have | | < 1 We might accomplish this by means of a differential equation, in the manner indicated in the case of leal variables (I, 183, 2d ed , 179, 1st ed ) The following method, which gives an application of 11, is more closely related to the method fol- lowed by Abel. We shall suppose a fixed and \\ < 1, a&d we shall study the properties of </> (w, 2) considered as a function of m. If m is a positive integer, the function evidently reduces to the polynomial (1 + z) m . If m and m' are any two values whatever of the parameter m, we have always (30) < (w, *) < (m f , ) = < (m -f m', z) In fact, let us multiply the two series < (m, #), <j> (w', ) by the ordi- nary rule The coefficient of # p in the product is equal to (31) m p + m^^m{ -f m p _ 2 ^2 + + ?Vi-i + K>> where we have set for abbreviation m (m 1) - (m k + 1) m * = ^ - -. - *. The proposed functional relation will be established if we show that the expression (31) is identical with the coefficient of # p in <f> (m + m f , #), that is, witK (m + m')^ We could easily verify directly the identity (32) (m -f m') p ~ but the computation is unnecessary if we notice that the relation (30) is always satisfied whenever m and m' are positive integers The two sides of the equation (32) are polynomials m m and m' which are equal whenever m and m 1 are positive integers, they are therefore identical. On the other hand, <(#&, z) can be expanded in a power series of increasing powers of m. In fact, if we carry out the indicated products, < (m, &) can be considered as the sum. of a double series I, 18] POWER SERIES WITH COMPLEX TERMS 41 <j>(m, z)=l + ~ g;-^:^2 + s -t P qp ... J. -tf O 1? /33\ ^ . ?M? 2 _ ra 3 a i . \ / o 9* "^ 771 8 - ^Z. P ~r" / * "^ i ~~ t " O i? i if we sum it by columns This double series is absolutely convergent [For, let \z\ = /> and \m\ = o- , if we replace each term by its absolute value, the sum of the terms of the new series included in the (p + l)th column is equal to o-Qr + 1) ( wliicli is the general teim of a convergent series We can therefore sum the double senes by rows, and we thus obtain for <t>(m, z) a development in power series From the relation (30) and the results established above ( 11), this series must be identical with that for e aim . Now for the coeffi- cient of m we have hence 1 (34) where the determination of the logarithm to be understood is that one which becomes zero when z = 0. We can again represent the last expression by (1 + z) m , but in order to know without ambiguity the value in question, it is convenient to make use of the expression Let m = p + v i , if r and a have the same meanings as in the preceding paragraph, we have gmLog (1+ a) ___ ^(l* + it) (log r + tar) = e ftl sr ~ ^[cos (pa + v logr) + i sin (pa + v logr)]. In conclusion, let us study the series on the circle of convergence. Let U n "be the absolute value of the general term for a point z on the circle The ratio of two consecutive terms of the senes of absolute values is equal to | (m w+l)/n|, that is, if m = n + w, to 42 ELEMENTS OF THE THEORY [I, 18 where the function $(n) remains finite when n increases indefinitely By a known rule for convergence (I, 163) tins senes is conveigent when /i + 1 > 1 and divergent in every other case The senes (29) is therefore absolutely con- vergent at all the points on the circle of convergence when AC is positive If fj, + 1 is negative or zero, the absolute value of the geneial teim never decreases, since the ratio U n +i/U n is nevei less than unity The senes w> diver- gent at all the points on the circle when /x =i 1. It remains to study the case where 1< /* =i Let us con&idei the series whose general term is U* , the ratio of two consecutive terms is equal to .*(")"!* i _-_ i i n 2 J and if we choose p laige enough so that p (/* + 1) > 1, this series will be conver- gent. It follows that J7, and consequently the absolute value of the geneial term l/ n , appi caches zero. This being the case, in the identity let us retain on each side only the terms of degree less than or equal to n , there remains the i elation ^ where S n and S^ indicate respectively tho sum of the fix at (n + 1) toims of 0(m, z) and of 0(m + 1, 2) If the ical pait of m lies between 1 and 0, the real part of m + 1 is positive Suppose | z \ = 1 , when the nuinbex n increases indefinitely, <8 approaches a limit, and the labt teim on the right appioachos zero , it follows that S n also approaches a limit, unless 1 + 2 = Theiefore, when 1< AC 35 0, the series is convergent at all the points on the circle of conver- gence, except at the point 2 = 1 IIT CONFORMAL RE PRESENTATION 19 Geometric interpretatxon of the derivative. Lot n = -V + Yl 1> a function of the complex variable , analytic within a closed curve C. We shall represent the value of n by the point whose coordinates aie JT, Y with respect to a system of rectangular axes. To simplify the following statements we shall suppose that the axes OX, Y are par- allel respectively to the axes Ox and Oy and ai ranged in the same order of rotation in the same plane or in a plane parallel to the plane trOy. When the point % describes the legion A bounded by tho closed curve C, the point u with the coordinates (-Y, F) describes in its plane a region A' , the relation u ==/() defines then a certain corre- spondence between the points of the two planes or of two portions of a plane On account of the relations which connect the derivatives of the functions X, F, it is clear that this correspondence should possess special properties. We shall now show that the angles are unchanged. I, COJSIFORMAL REPRESENTATION 43 Let # and ^ be two neighboring points of the region A, and u and Wj the coi responding points of the region A'. By the original defini- tion of the derivative the quotient (2^ u)/(^ z) has for its limit f(z) when the absolute value of x & approaches zeio in any manner whatever Suppose that the point l approaches the point 2 along a cuive C, whose tangent at the point & makes an angle a with the paiallel to the direction Ox , the point ^ will itself de- scribe a cuive C' passing through u Let us discard the case in which f'(z) is zeio, and let p and <D be the absolute value and the angle of f(z) respectively Likewise let r and r' be the distances z^ and uu v a' the angle which the direction ^ makes with the parallel &x? to Ox, and ft' the angle which the direction uu^ makes with the paiallel uX' to OX The absolute value of the quotient \0' I) a FIG 96 (7 a u)/( t z) is equal to i\/r, and the angle of the quotient is equal to ft' a'. We have then the two relations (35) liin (p a 1 } = 2 kir Let us consider only the second of these relations. We may sup- pose k = 0, since a change in k simply causes an increase in the angle w by a multiple of 2?r. When the point ^ approaches the point & along the curve C, a 1 approaches the limit a, ft' approaches a limit ft, and we have ft = a + That is to say, in 01 der to obtain tJie direction of the tangent to the curve described by the point ir, it suffices to turn the direction of the tangent to the curve described by & through a constant angle o> It is naturally understood in this statement that those dnections of the two tangents are made to coriespond which correspond to the same sense of motion of the points & and u. Let D be another curve of the plane xOy passing through the point Zj and let D f be the corresponding curve of the plane XOY If the letters y and 8 denote zespectively the angles which the corresponding 44 ELEMENTS OF THE THEORY [I, 19 directions of the tangents to these two curves make with zx' and uX' (Figs, 9 a and 9 #), we have /3 = #-f<f>, S = y + o>, and consequently 8 /3 = y a The curves C' and D* cut each other in the same angle as the curves C and D Moreovei, we see that the sense of rotation is pieserved. It should be noticed that if /'(#) = 0, the demonstration no longei applies If, in particular, we considei, in one of the two planes xOy or XOY, two families of orthogonal curves, the corresponding curves in the other plane also will form two families of orthogonal cuives For example, the two families of curves X = C, Y = C', and the two families of curves (36) !/(*)!= C, angle /(*)=C" form orthogonal nets in the plane x 0y, for the corresponding curves in the plane XOY are, in the first case, two systems of paiallels to the axes of coordinates, and, in the other, circles having the origin for center and straight lines proceeding from the origin Example 1 Let z f = a?*, wheie a is a real positive numbei Indicating by r and 6 the polar coordinates of 2, and by r' and Q' the polai cooidmates of 2', the preceding relation becomes equivalent to the two lelations r' = ? a , 6' = a& "We pass then from the point z to the point 2' by raising the ladius vectoi to the powei a and by multiplying the angle by a The angles are pieseived, ex- cept those which have their vciticos at the ongin, and these aie multiplied by the constant factoi a Example 2 Let us consider the general linear transformation - = where a, 5, c, d are any constants whatever In ceitam paiticular cases it is easily seen how to pass from the point z to the point z'. Take for example the transformation zf = z + b , let 2 = to + yi"> %' == *' -f y't, b = a + /& , the picced- ing relation gives x' = x -f a, y' = 2/ + ft which shows that wo pass fiom the point z to the point z f by a translation Let now zf = az , if p and w indicate the absolute value and angle of a lospec- tively, then we have r f = /or, 6' = w + ^ Hence we pass f lorn the point 2 to the pomt^ by multiplying the radius vector by the constant factoi p and then turning this new radius vector through a constant angle w We obtain then the transfor- mation defined "by the formula z' = az by combining an expansion with a rotation Finally, let us consider the relation where r, 0, r x , 0' have the same meanings as above We must have rr' = 1, B + Q' = The product of the radu vectores is therefore equal to unity, while I, 20] CONFORMAL REPRESENTATION 45 the polar angles aie equal and of opposite signs Given a circle O with center A and radius E, we shall use the expression mv&sion with respect to the given circle to denote the transformation by which the polar angle is unchanged but the radius vector of the new point is R 2 /r We obtain then the transformation defined by the relation z'z = 1 by carrying out first an inversion with respect to a circle of unit radius and with the origin as centei, and then taking the sym- metric point to the point obtained with respect to the axis Ox The most general transfoimation of the foim (37) can be obtained by com- bining the transformations which we have just studied If c = 0, we can replace the transformation (37) by the succession of transformations a , . b *, = -*, z' = z 1 + - If c is not zero, we can carry out the indicated division and write , __ a be ad Z ~" and the transformation can be leplaced by the succession of tiansformations *! = * + -* 2 2 = c 2 *i, 3 = - c z z z 4 = (6c ad) z s , z' = z 4 -f - c All these special transformations leave the angles and the sense of rotation unchanged, and change circles into circles Hence the same thing is then tiue of the general transformation (37), which is therefore often called a circular transformation In the above statement straight lines should be regarded as circles with infinite radii Example 3 Let Z 7 = (z - etf*i (z - e^ (z - %,)*, where e 1? e 2 , , % are any quantities whatever, and where the exponents %, m 2 , , nip are any real numbers, positive or negative Let If, J? t , jS? 2 -^P be the points which represent the quantities 2, e v e^ - , p , let also r 3 , r 2 , - 7 r p denote the distances JfJBfj, 3O? 2 , , ME P and a , 2 , , P the angles which IS^ Jf, jE? 2 Jf, , E P M make with the parallels to Ox The absolute value and the angle of z' are respectively r^rj** - * r/>v and m^ + m 2 2 + Then the two families of curves form an orthogonal system When the exponents m t , m a , , ra p are rational numbers, all the curves are algebraic If, for example, p = 2, m t = m 2 = 1, one of the families is composed of Cassmian ovals with two foci, and the second family is a system of equilateral hyperbolas 20. Confonnal transformations in general. The examination of the converse of the proposition which we have just established leads us to treat a more general problem Two surfaces, 2, S', being given, let us set up between them any point-to-point correspondence whatever 46 ELEMENTS OF THE THEORY LI, 20 (except for certain broad restrictions winch will be made latei), and let us examine tlie cases in which the angles are unaltered in that transformation Let x, y, * be the rectangular coordinates of a point of S, and let x\ y f , ' be the lectangular cooidmates of a point of 3' We shall suppose the six coordinates x, y, z, x 1 , y\ r expiessed as functions of two variable parameters it, v in such a way that corresponding points of the two surfaces coi respond to the same pair of values of the parameters u, v (38) s'-U' = . '(*, ), Moreover, we shall suppose that the functions /, <, , together with their paitial derivatives of the first order, aie continuous when the points (SB, y, s) and (a;', y\ z 1 ) remain in ceitain legions of the two surfaces 2 and 5' We shall employ the usual notations (I, 131) (39) -<&' '-' *-><$$ 2 Fdu dv JB'du* + ZF'du di) + G'dv*. Let C and D (Figs 10 a and 10 J) be two cuives on the stufacc ^, passing through a point m of that suiface, and C' and />' the corre- sponding curves on the surface 5' passing through the point w,' FIG 10 a . 106 Along the curve C the parameters u t v are functions of a single auxiliary variable t, and we shall indicate their differentials by du, and dv Likewise, along D, u and v are functions of a variable ', and we shall denote their diiferentials here by 8u and Sw In general, we shall distinguish by the letters d and 8 the differentials relative to a displacement on the curve C and to one on the curve D. The I, 20] CONFORMAL REPRESENTATION 47 following total diffeientials are pioportional to the direction cosines of the tangent to the curve C, , dx , dx _ _ Bt/ T a?/ _ 7 da . dx J dx = du + -%-dv, dy = ^-du + -^dv, dz du + -z-dv, on cv y cu ov cu ov and the following are proportional to the direction cosines of the tangent to the curve D, , ov ' du tic Let <o be the angle between the tangents to the two curves C and D The value of cos o> is given by the expiession dx Bx + d y 8y 4- dz $z cos w - J J -- ^dx 2 + dif + which can be written, making use of the notation (39), in the form Edu &/ -f F(du Sv + du 8?Q + Gdv $v (40) cos o> = + 2Fdu dv + Gdv z VE Sit? -f 2 FStc v -f G Sv 3 If we let o) f denote the angle between the tangents to the two curves C' and Z> f , we have also ,. , E'du 8z/ + F'(du S?; -f dv Bit} + G'clv Bv (41) COS a/ = / - ^ y In order that the transformation considered shall not change the value of the angles, it is necessary that cos o>' = cos o>, whatever du, dv, Sit, Sv may be The two sides of the equality cos 2 a/ = cos 2 CD are rational functions of the latios Sv/Sit, dv/du, and these functions must be equal whatever the values of these ratios Hence the corre- sponding coefficients of the twp fractions must be proportional , that is, we must have where X is any function whatever of the parameters it, v These conditions are evidently also sufficient, for cos o>, for example, is a homogeneous function of E, F, G, of degree zero The conditions (42) can be replaced by a single relation ds = X% 2 , 01 (43) ds' = \ds. 48 ELEMENTS OF THE THEORY [I, 20 This relation states that the ratio of two coi responding infinitesimal arcs approach a limit independent of du and of dv 9 when these two arcs approach zero This condition makes the reasoning almost intuitive Foi, let abc be an infinitesimal triangle on the first surface, and a'W the coi responding triangle on the second surface Imagine these two curvilineai triangles replaced by rectilinear triangles that approximate them Since the ratios a'V/ab, a'c'/ac, !>'c'/t>c approach the same limit X(M, v), these two triangles approach similarity and the corresponding angles appioaeh equality We see that any two corresponding infinitesimal figures on the two surfaces can be consideied as similar, since the lengths of the aics are pioportional and the angles equal , it is on this account that the term conformal representation is often given to every correspond- ence which does not alter the angles Given two surfaces 2, S' and a definite relation which establishes a point-to-point conespondence between these two surfaces, we can always determine whether the conditions (42) are satisfied or not, and therefore whether we have a conformal lepresentation of one of the surfaces on the other But we may consider other problems For example, given the sur- faces S and S', we may propose the problem of determining all the coirespondences between the points of the two surfaces which pre- seive the angles Suppose that the coordinates (x, y, z) of a point of 5 are expressed as functions of two parameters (, v), and that the coordinates (x', y', *) of a point of S' are expressed as functions of two other parameters (u' 9 v') Let ?, ds a = E f du'* + 2 F f du 1 dv' be the expressions for the squares of the lineai elements The prob- lem in question amounts to this To find two functions u' = TT^M, -y), v 1 = 7T 2 (w, v) suck that we have identically d'7r i + G l d-rr\ = \\E du* + 2 F du, dv X being any function of the variables u, v The geneial theory of dif- ferential equations shows that this problem always admits an infinite number of solutions , we shall consider only certain special cases 21. Conformal representation of one plane on another plane. Every correspondence between the points of two planes is defined by relations such as (44) X = P(x,y), F=Q(^y), I, 21] COISTFORMAL REPRESENTATION 49 where the two planes are referred to systems of rectangular coordi- nates (x , ?/) and (A", F) From what we have just seen, in order that this tiansformation shall preseive the angles, it is necessaiy and sufficient that we have <LY* + dY* = \\dx* + df), where X is any function whatever of x 9 y independent of the differ- entials Developing the differentials dX, <#Fand comparing the two sides, we find that the two functions P(x, y) and Q(r, y) must satisfy the two relations dx dy dx dy The partial derivatives dP/dy, dQ/dy cannot both be zero, for the first of the relations (45) would give also dQ/dx = dP/dx = 0, and the functions P and Q would be constants. Consequently we can write according to the last relation, <9P_- 3Q 2Q__ &P dx dy dx dy where /-i is an auxiliary unknown. Putting these values in the first condition (45), it becomes and from it we derive the result ;LC = 1. We must then have either dP dQ, dP 3Q (4b) 5~" == V -.= ^ ' ox uy oy ox or (47) ^^^^Q, ^ = 5. ^ ' dx dy dy dx The first set of conditions state that P + Qi is an analytic func- tion of x + yi As for the second set, we can reduce it to the first by changing Q to Q, that is, by taking the figure symmetric to the transformed figure with respect to the axis OX Thus we see, finally, that to every conformal representation of a plane on a plane there corresponds a solution of the system (46), and consequently an analytic function If we suppose the axes OX and OY parallel re- spectively to the axes Ox and Oy, the sense of rotation of the angles is preserved or not, according as the functions P and Q satisfy the relations (46) or (47) 50 ELEMENTS OF THE THEORY [I, 22 22. Riemann's theorem Given in the plane of the vanable * a region A bounded by a single curve (or simple boundary), and in the plane of the vari- able u a circle C, Riemann proved that there exists an analytic function u = /(*), analytic in the legion A, such that to each point of the legion A conesponds a point of the cncle, and that, concisely, to a point of the oiiole conesponds one and only one point of A The function /() depends also upon thiee aibitiaiy leal constants, which we can dispose of in such a way that the center ot the cncle coiiesponds, to a given point of the region A, while an aibitianly chosen point on the cncuinfeience corresponds to a given point of the boundaiy of A We shall not give here the demonstiation of this theorem, of which we shall indicate only some examples We shall point out only that the circle can be replaced by a half -plane Thus, let us suppose that, in the plane of u, the circumference passes through the origin , the transfoimation u' = 1/u lepJaces that circumference by a straight line, and the circle itself by the poition of the w'-plane situated on one side ot the stiaight line extended indefinitely in both directions Example 1 Let u = z 1 /*, where a is real and positive Consider the portion A of the plane included between the direction Ox and a ray through the origin making an angle of cm with Ox (a ^ 2). Let z = re*, u = R& , we have B-4 w =- ~~ ' "" at When the point z describes the portion A of the plane, r varies from to f oo and 9 from to air , hence R varies fiom to + oo and w fiom to it y ' i ! it FIG. 11 The point u therefore describes the half -plane situated above the axis OX, and to a point of that half -plane corresponds only one point of -4, for we have, inversely, r = It", Q = a<a Let us next take the portion B of the #-plane bounded by two arcs of circles which intersect Let 3 , z l be the points of intersection , if we carry out first the transformation the region B goes over into a portion A of the emplane included between two rays from the origin, for along the arc of a -circle passing through the points I, 22] CONFORMAL REPRESENTATION 51 2 , z lt the angle of (z z )/(z 2 X ) lemams constant Applying now the pre- ceding tianstormation u = (z') l /, we see that the function enables us to lealize the conformal lepiesentation of the region B on a half- plane by suitably choosing a Example 2 Let u = cos 2 Let us cause z to describe the infinite half -strip JB, or AOBA' (Fig 11), defined by the inequalities ^ x ^ IT, y a 0, and let us examine the legion described by the point u = X + Yi We have here ( 12) (48) When a; varies fiom to ir, F is always negative and the point u remains in the half -plane below the axis X'OX Hence, to eveiy point of the region E corresponds a point of the u half-plane, and when the point z is on the bound- aiy of JR, we have T = 0, for one of the two factois sin x, 01 (& e-v)/2 is zeio Conversely, to every point of the u half -plane below OX corresponds one and only one point of the strip E in the s-plane In fact, if z' is a root of the equa- tion u cos 2, all the other roots are included in the expression 2 Tcir z* If the coefficient of ^ in z f is positive, there cannot be but one of these points in the strip E, for all the points 2 kv z' are below Ox There is always one of the points 2 for + zf situated in JR, for there is always one of these points whose abscissa lies between and 2 TT That abscissa cannot be included between TT and 2 TT, for the corresponding value of Y would then be positive The point is therefore located in E It is easily seen fiom the formulae (48) that when the point z describes the portion of a paiallel to Ox in JR, the point u describes half of an ellipse When the point z describes a parallel to Oy, the point u describes a half-branch of a hyperbola All these conies have as foci the points C, G" of the axis O.X, with the abscissas + 1 and I Example 3 Let vz (49) where a is real and positive In order that \u\ shall be less than unity, it is easy to show that it is necessary and sufficient, that cos [(7r^)/(2 a)] > If y vanes from a to + a, we see that to the infinite stup included between the two straight lines y = a, y = + a corresponds in the w-plane the circle C described about the origin as center with unit radius Conversely, to every point of this circle corresponds one and only one point of the infinite stnp, for the values of z which correspond to a given value of u form an arithmetical pro- gression with the constant difference of 4 ai Hence there cannot be more than one value of z in the stnp considered Moreover, there is always one of these roots in which the coefficient of i lies between a and 3 a, and that coefficient cannot he between a and 3 a, for the corresponding value of | u | would then be greater than umty 52 ELEMENTS OF THE THEOEY [I, 23 23. Geographic maps. To make a confoimal map of a surface means to make the points of the surface coi respond to those of a plane in such a way that the angles are unalteied Suppose that the cooidmates of a point of the surface S undei consideration be ex- pressed as functions of two variable parameters (u, v), and let -f- 2 Fdu dv + G dv 2 be the square of the linear element for this surface Let (a, ft) be the lectangular coordinates of the point of the plane P which cor- responds to the point (u, v) of the surface The problem here is to find two functions w = 7^(0,0), * = ir fl (*,) of such a nature that we have identically where A. is any function whatever of <x y fi not containing the differ- entials This problem admits an infinite number of solutions, which can all be deduced from one of them by means of the conformal tiansformations, already studied, of one plane on another Suppose that we actually have at the same time df = \(d<P + dfP), ds* = X f (da* + dp*) ; then we shall also have da* + df? = ~ (<fo a + dp*), A so that a + pi, or a pi, will be an analytic function of a* + p'i The converse is evident Example 1 Mercator's projection We can always make a map of a surface of revolution in such a way that the meridians and the paral- lels of latitude correspond to the parallels to the axes of coordinates Thus, let be the coordinates of a point of a surface of revolution about the axis 0& , we have which can be written if we set I, 23] CONFORMAL REPRESENTATION 53 In the case of a sphere of radius R we can write the coordinates in the form x = R sin cos <, y = R sin 6 sin <, & = J2 cos 0, and we shall set We obtaui thus what is called Mercator's projection, in which the meridians are represented by parallels to the axis OY, and the paral- lels of latitude by segments of straight lines parallel to OX. To obtain the whole surface of the sphere it is sufficient to let <f> vary from to 2 TT, and from to TT , then A" varies from to 2 TT and Y from oo to + oo The map has then the appearance of an infinite strip of breadth 2 TT The curves on the surface of the sphere which cut the meridians at a constant angle are called loxodromio curves or rhumb lines, and are represented on the map by straight lines. Example 2. Stereo graphic projection. Again, we may write the square of the linear element of the sphere in the form or ds* = 4 cos 4 1 (dp* + p^o 2 ), if we set Q p = jR tan jr ? co =$. But efy> a + /) 2 ^w 2 represents the square of the linear element of the plane in polar coordinates (p, o>) ; hence it is sufficient, in order to obtain a conf ormal representation of the sphere, to make a point of the plane with polar coordinates (p, o>) correspond to the point (0, <) of the surface of the sphere It is seen immediately, on drawing the figure, that p and <o are the polar coordinates of the stereograpluc projection of the point (0, <) of the sphere on the plane of the equator, the center of projection being one of the poles * * The center of prelection is the south pole if 6 is measured from the north pole to the radius Using the north pole as the center of projection, the point (IP/p, w), symmetric to the first point (see Ex 17, p 58), would be obtained TRANS 54 ELEMENTS OF THE THEORY [I, 23 Example 3 Map of an anchor ring Consider the anchoi ring generated by the revolution of a circle of ladius R about an axis situated in its own plane at a distance a from its centei, wheie a > R Taking the axis of i evolution for the axis of 3, and the median plane of the anchor ring foi the icy-plane, we can wnte the coordinates of a point of the surface in the foim x = (a 4- E cos 0) cos 0, y = (a + It cos0)sm0, z = #sm0, and it is sufficient to let 8 and $ vaiy from ir to + if From these formulae we deduce and, to obtain a map of the suiface, we may set where Thus the total surface of the anchor ring corresponds point by point to that of a rectangle whose sides are 2 TT and 2 we/ Vl e 2 34. Isothermal curves Let U(x, y) be a solution of Laplace's equation the curves represented by the equation (60) U(x, y) = C, where C7 is an aibitraiy constant, form a family of isothermal curves With every solution &(x, y) of Laplace's equation we can associate another solution, "7(05, y), such that U + Viis an analytic function of x + yi The relations 8CT = 3F ^ = _.Z d& dy dy dx show that the two families of isothermal curves 0>, y) = 0, F(fc, y) = C' are orthogonal, for the slopes of the tangents to the two curves C and C' are respeomely _S_8U > _8F_F 3aj 5y 2x dy Thus the orthogonal trajectories of a family of isothermal curves form another family of isothermal curves We obtain all the conjugate systems of isothermal curves by considering all analytic functions f(z) and taking the curves for which the real part of f(z) and the coefficient of % have constant values The curves for which the absolute value E and the angle O of f(z) remain constant also form two conjugate isothermal systems , for the real part of the analytic function Log [/()] is log 22, and the coefficient of i is Q Likewise we obtain conjugate isothermal systems by considering the curves described by the point whose coordinates are JT, F, where f(z) = X + Yi, when I, 24] COHFORMAL REPRESENTATIONS 55 we give to x and y constant values. This is seen by regarding x -f- yi as an analytic function of J5T+ Yi More generally, every tiansfoiination of the points of one plane on the other, which preserves the angles, changes one family of isothermal curves into a new family of isothermal curves Let be equations defining a transformation which preserves angles, and letF(o;', y') be the result obtained on substituting p (x', y') and q (x', tf) for x and y in 27(0;, y) The proof consists in showing that F(JC', 2/0 is a solution of Laplace's equation, provided that U(x, y) is a solution The verification of this fact does not offer any difficulty (see Vol I, Chap III, Ex 8, 2d ed , Chap II, Ex 9, 1st ed ), but the theorem can be established without any calculation Thus, we can sup- pose that the functions p (#', y^) and q (x', ]/) satisfy the relations for a symmetric transformation evidently changes a family of isothermal curves into a new family of isothermal curves. The function x + yi = p + qi is then an analytic function of zf = tf + y% and, after the substitution, IT 4- Vi also becomes an analytic function F(x', y*) + v$ (&', y') of the same variable z' ( 5) Hence the two families of curves =0, $(',20 = 0' give a new orthogonal net f oimed by two corrugate isothermal families. Ifoi example, concentric cucles and the rays from the center form two con- jugate isothermal families, as we see at once by considering the analytic func- tion Log z Carrying out an inversion, we have the result that the circles passing through two fixed points also form an isothermal system. The conjugate system is also composed of circles Likewise, conf ocal ellipses form an isothermal system. Indeed, we have seen above that the point u = cos z descnbes conf ocal ellipses when the point z is made to descnbe parallels to the &xis Ox ( 22) The conjugate system is made up of conf ocal and orthogonal hyperbolas Note In order that a family of curves represented by an equation P (, y) = C may be isothermal, it is not necessary that the function P (, y) be a solution of Laplace' s equation Indeed, these curves are represented also by the equation 0[P(ai, y)] = 0, whatever be the function <p , hence it is sufficient to take for the function a form such that U(x, y) = 4>(P) satisfies Laplace's equation Making the calculation, we find that we must have hence it is necessary that the quotient 3 2 P gP depend only on P, and if that condition is satisfied, the function can be obtained by two quadratures. 56 ELEMENTS OF THE THEORY [I, Exs EXERCISES 1 Determine the analytic function f(z) = X + Ti whose real part X is equal to 2sm2s Consider the same question, given that -"+ T is equal to the preceding function 2 Let (m, p) = be the tangential equation of a real algebraic curve, that is to say, the condition that the straight line y = ma; -f P be tangent to that curve The roots of the equation (&, zi) = are the real foci of the curve 3 If p and q are two integers prime to each other, the two expressions (Vz) p and VZP are equivalent What happens when p and q have a greatest common divisor d > I ? 4 Pmd the absolute value and the angle of e^ + J^ by considering it as the limit of the polynomial [1 + (x + yi)/m\ m when the integer m increases indefinitely 5. Piove the formulae /n + 1 A cos a + cos(a + 6) -f + cos(a + rib) = cos (a + ) , sm(-^ X 2/ I n5 sina + am (a + 6) + - + sm (a + rib) = - sin la + y 6. What is the final value of arc sin z when the variable z describes the seg- ment of a straight line from the origin to the point 1 + *, if the initial value of arc sm z is taken as 9 7. Prove the continuity of a power series by means of the formula (12) ( 8) /( + *)-/() = A/! (2) + J5/,<) + + 5/( Z > + [Take a suitable dominant function for the series of the right-hand side ] 8 Calculate the integrals i x m QOX C0 g ftp ^ Cym 000; SIn fo> $, fctn(cc a) ctn(x~ 6) - ctn(sc Z)dcc 9 Given in the plane xOy a closed curve (7 having any number whatever of double points and described in a determined sense, a numerical coefficient is assigned to each region of the plane determined by the curve according to the rule of Volume I ( 97, 3d ed , 96, 1st ed) Thus, let .K, B' be two contiguous regions separated by the arc ab of the curve described in the sense of a to 6 , the coeffi- cient of the region to the left is greater by unity than the coefficient of the region to the right, and the region exterior to the curve has the coefficient I,Exs] EXERCISES 57 Let ZQ be a point taken in one of the regions and N the corresponding coeffi- cient Prove that 2Nv represents the variation of the angle of z z ^hen the point z describes the curve C m the sense chosen 10 By studying the development of Log[(l+ z)/(l 2)] on the circle of convergence, prove that the sum of the series smfl sm80 sm50 sin (2 u 1 3 5 2n+l is equal to ?r/4, according as sm 6 > (Cf Vol I, 204, 2d ed , 198, 1st ed ) 11 Study the curves described by the point Z = z z when the point z describes a straight line or a circle 12 The relation 2Z = z + c*/z effects the conformal representation of the region inclosed between two conf ocal ellipses on the ring-shaped region bounded by two concentric circles [Take, for example, z = Z -f Vz 2 c 2 , make m the Z-plane a straight-line cut ( c, c), and choose for the radical a positive value when Z is real and greater than c ] 13. Every circular transformation z' = (az + b)/(cz + d) can be obtained by the combination of an eoen number of inversions Prove also the converse 14 Eveiy transformation defined by the relation zf = (az + b)/(cz + d), where Z Q indicates the conjugate of z, results from an odd number of inversions Prove also the converse 15. Fuchsian transformations. Every linear transformation ( 19, Ex 2) z' = (az + b)/(cz + d), where a, 6, c, d are real numbers satisfying the relation ad 6c = 1, is called a Fuchsian transformation Such a transformation sets up a correspondence such that to every point z situated above Ox corresponds a point z' situated on the same side of Ox' The two definite integrals / dxdy -- are invariants with respect to all these transformations The preceding transformation has two double points which correspond to the roots or, of the equation as 2 + (d a)2 6 = If a and are real and distinct, we can write the equation z' = (az + V)/(cz + d) m the equivalent form where k is real Such a transformation is called hyperbolic. If cc. and $ are conjugate imaginan.es, we can write the equation where is real Such a transformation is called elliptic If j8 = a, we can write where a and k are real Such a transformation is called parabola. 58 ELEMENTS OF THE THEORY [I,Exs 16 Let z' =/() be a Fuchsian transformation Put Prove that all the points z,z 1 ,z 2 , , z n are on the circumf eience of a circle. Does the point z n approach a limiting position as n increases indefinitely ? 17. Given a circle C with the center O and radius JR, two points 3f, JfcP situated on a ray fiom the center are said to be symmetric with respect to that circle if OM x OM' = R* Let now C, C' be two cncles in the same plane and M any point whatevei in that plane Take the point Jlfj symmetric to M with respect to the circle (7, then the point M { symmetric to M with respect to C", then the point Jf 2 sym- metric to M{ with respect to C, and so on forever Study the distribution of the points .M^, -M"i, Jf 2 , Jj, " 18. Find the analytic function Z=f(z) which enables us to pass from Mercator's projection to the stereogiaphic projection 19* All the isothermal families composed of circles are made up of circles passing through two fixed points, distinct 01 coincident, real or imaginary [Setting z = x 4- 2/1, Z Q = x yi, the equation of a family of circles depending upon a single parameter X may be written in the form ZZ Q + az + bz + c = 0, where a, 6, c are functions of the parameter X In order that this family be isothermal, it is necessary that d z \/dzdz = Making the calculation, the theorem stated is proved ] 20*. If |g| < 1, we have the identity [EULER ] [In order to prove this, transform the infinite product on the left into an infinite product with two indices by putting in the first row the factors 1 + g, 1 + g 2 , 1 + g 4 , , l + g 2 ", , m the second row the factors 1 + g s , 1+9 6 , i 1 + (g s ) 2n , - ; and then apply the formula (16) of the text ] 21. Develop in powers of z the infinite products F(z) = (1 + xz) (1 + x*z) (1 + xz) , *(z) = (1 -f xz) (1 -f a^) (l + x *+iz) .... [It is possible, for example, to make use of the relation F(xz) (1 + xz) = F(z), *(x*z) (I + xz) = 22*. Supposing \x\ < 1, prove Euler's formula (See J. BBBTKAND, (7afcuZ d^rewiicZ, p 328 ) I, Exs ] EXERCISES 59 23* Given a sphere of unit radius, the stereographic projection of that sphere is made on the plane of the equatoi, the center of projection being one of the poles To a point M of the sphere is made to correspond the complex number s = x + yi, where x and y are the lectangular coordinates of the projection m of M with lespect to two rectangular axes of the plane of the equator, the origin being the center of the sphere To two diametrically opposite points of the sphere coriespond two complex numbers, s, l/s , where s is the conjugate imaginary to 8 Every linear transformation of the form (A) where p<x + 1 = 0, defines a rotation of the sphere about a diameter. To groups of rotations which make a regular polyhedron coincide with itself correspond the groups of finite order of linear substitutions of the form (A). (See KLEIN, Das Ikosaeder ) CHAPTER II THE GENERAL THEORY OF ANALYTIC FUNCTIONS ACCORDING TO CAUCHY I DEFINITE INTEGRALS TAKEN BETWEEN IMAGINARY LIMITS 25. Definitions and general principles. The results presented in the preceding chapter are independent of the work of Cauchy and, for the most part, prior to that woik We shall now make a system- atic study of analytic functions, and determine the logical conse- quences of the definition of such functions Let us recall that a function f(z) is analytic in a region A . 1) if to every point taken in the region A corresponds a definite value of f(z) ; 2) if that value varies continuously with & ; 3) if for every point taken in A. the quotient / + &)-/() h approaches a limit f(z) when the absolute value of h approaches zero. The consideration of definite integrals, when the variable passes through a succession of complex values, is due to Cauchy * ; it was the origin of new and fruitful methods. Let f() be a continuous function of 2 along the curve A MB (Fig. 12) Let us mark off on this curve a certain number of points of division , v 3 2 , , s n _ 1? 2', which follow each other in the order of increasing indices when the arc is traversed from A to B, the points & and #' coinciding with the extremities A and B Let us take next a second series of points 1? 2 , -, n on the arc AB, the point & being situated on the arc *|._i* t and let us consider the sum +/k) (** - **-0 + +/( <*' - *-0 When the number of points of division 1? , n-1 increases indefi- nitely in such a way that the absolute values of all the differences * Memoire sur les integrates defimes, prises entre des hrmtes ^mag^na^res, 1825 This memoir is reprinted m Volumes VII and VIII of the Bulletin des Sciences math& matiques (1st series) 60 n, 25] DEFINITE INTEGRALS 61 z i ~~ *o> ^2 ~~ *i> beeome and remain smaller than any positive number arbitrarily chosen, the sum S approaches a limit, which is called the definite integial of /() taken along AMB and which is represented by the symbol L (.AMB) To prove this, let us separate the real part and the coefficient of i in S, and let us set FIG 12 where X and Y are continuous functions along AMB. Uniting the similar terms, we can write the sum S in the form l - fl >+" ] When the number of divisions increases indefinitely, the sum of the terms in the same row has for its limit a line integral taken along AMJB, and the limit of S is equal to the sum of four line integrals:* f /()<& = f (Xdx - Ydy) 4- i T J(AMS) JuilB) JUMB) * In order to avoid useless complications in the proofs, we suppose that the coor- dinates x t y of a point of the arc AMB are continuous functions x= $ (<), y ~ $ (2) of a parameter t, which have only a finite number of maxima and minima "between A and B We can then hreak up the path of integration into a finite numoer of arcs which are each i epresented by an equation ol the form y*=F(ti), the function F being continuous between the corresponding limits , or into a finite number of arcs which are each represented by an equation of the form fc= G (y) There is no disadvantage in making this hypothesis, for in all the applications there is always a certain amount of freedom in the choice of the path of integration Moreover, it would suffice to suppose that (2) and V ($) are functions of limited vanation We have seen that in this case the curve AMB is then rectifiable (I, ftns , 73, 82, 95, 2d ed ). 62 THE GENERAL CAUCHY THEORY [H,25 From the definition it results immediately that f f(*)d*+C /(*)<fe = J(AMB) J(BMA) It is often important to know an upper bound for the absolute value of an integral Let s be the length of the arc AM, L the length of the arc AB, s L _ l9 s^, ^ the lengths of the arcs A^_ ly Az L , A& of the path of integration Setting F(s) = |/() |, we have |/(&)(*i - **-i) I = F (*d K - *i-i| = F (*d ( s i - **-0 for | t ^_ 1 | represents the length of the chord, and S L S L _ I the length of the arc. Hence the absolute value of S is less than or at most equal to the sum ^F(a- k )(s L s*,_i) , whence, passing to the limit, we find , r r z I /(.)& S / F(s)ds. I J(AMB) */0 Let M be an upper bound for the absolute value of /() along the curve AB. It is clear that the absolute value of the integral on the right is less than ML, and we have, a fortiori, jL f(*)d* <ML. 26. Change of variables. Let us consider the case that occurs fre- quently in applications, in which the coordinates #, y of a point of the arc AB are continuous functions of a variable parameter t, x = <f> (), y = if/ (), possessing continuous derivatives <' (t), ij/' (t) ; and let us suppose that the point (x, y) describes the path of integra- tion from A to B as t varies from a to ft Let P() and Q(f) be the functions of t obtained by substituting <() and i^(tf), respectively, for x and y in X and F By the formula established for line integrals (I, 95, 2d ed.; 93, 1st ed.) we have X/0 Xdx Ydy = I 15) Jet Xr^ JS"^ + Ydx =s I [P(#) i/r'() + Q(#) ^ f (#)]^. UB) Jet Adding these two relations, after having multiplied tlie two sides of the second by *, we obtain (1) I f()dz= C [P(tf) */U,B) */* II, 26] DEFINITE INTEGRALS 63 This is precisely the result obtained by applying to the integral ff(z) dz the formula established for definite integrals in the case of real functions of leal variables, that is, in order to calculate the integral ff(&)dz we need only substitute <j>(f) + i\//(t) for & and [>'(*) + fy f (*)]<** for d * m/(s)d* The evaluation of ff(*)d* is thus reduced to the evaluation of two ordinary definite integrals. If the path A MB is composed of several pieces of distinct curves, the formula should be applied to each of these pieces separately. Let us consider, for example, the definite integral r +1 dz J-i We cannot integrate along the axis of reals, since the function to be integrated becomes infinite for # = 0, but we can follow any path whatever which does not pass thiough the origin Let & describe a semicircle of unit radius about the origin as center. This path is given by setting % == e tl and letting t vary from TT to 0. Then the integral takes the form /* l dz r Q r Q r Q ~= I ie-*dt=*il co8td+ I sm<& = 2. This is precisely the result that would be obtained by substituting the limits of integration directly in the primitive function 1/z according to the fundamental formula of the integral calculus (I, 78, 2d ed , 76, 1st ei). More generally, let z = <f> (u) be a continuous function of a new complex variable u = + i\i such that, when u describes in its plane a path C2VD, the variable describes the curve AMB To the points of division of the curve AMB correspond on the curve GND the points of division u , Uj, w 2 , - , uj._i, ui , ,u' If the function <j> (u) possesses a derivative <f>'(u) along the curve CND y we can write *L - *-! where e^ approaches zero when KI approaches U L ~I along the curve CND Taking ^t i = ^jt i an< i replacing z% ZLI by the expression derived from the preceding equality, the sum $, considered above, becomes 8 = A The first part of the right-hand side has for its limit the definite integral J(C2a>) 64 THE GENERAL CAUCHY THEORY [n, 26 As for the remaining term, its absolute value is smaller than t\ML', where 17 is a positive number greater than each of the absolute values | e |and where Z' is the length of the curve CND If the points of division can be taken so close that all the absolute values ] ej, | will be less than an arbitrarily chosen positive num- ber, the remaining term will approach zero, and the general formula for the change of variable will be (2) C f(z) dz = f f[<t> (u)] <t>'(u) du ^ ' J(AMB) J(CND) This formula is always applicable when (u) is an analytic function , in fact, it will be shown later that the derivative of an analytic function is also an analytic function* (see 34) 27. The formulae of Weierstrass and Darboux. The proof of the law of the mean f 01 integrals (I, 76, 2d ed , 74, 1st ed ) rests upon certain inequalities which cease to have a precise meaning when applied to complex quantities Weierstrass and Darboux, however, have obtained some interesting results in this connection by con- sidering integrals taken along a segment of the axis of reals We have seen above that the case of any path whatever can be reduced to this particular ease, provided certain mild restrictions are placed upon the path of integration. Let / be a definite integral of the following form . r =* f Ja * If this property is admitted, the following proposition can easily be proved Letf(z) be an analytic function in a finite region A of the plane For every pos^ tive number e another positive number ij can be found such that r (*)!<*, when z and z + h are two points of A whose distance from each other \h\is less than t\ For, let/ (2) = P (a;, y) + iQ (aj, y) , h = A + 1 Ay From the calculation made m 3, to find the conditions for the existence of a unique derivative, we can write , , _ t J(Z) ~ ^ [P' y (x + As, y + 0Ay) - P' y (a, y)] Ay ASB + i&y + Since the derivatives P^, Py , Q& Qy are continuous m the region A, we can find a num- ber 17 such that the absolute values of the coefficients of A* and of Ay are less than e/4, when VAi; 2 + Ay 2 is less than ij Hence the inequality written down above will be satisfied if we have ( h | < ij This being the case, if the function <f> (u) is analytic in the region A, all the absolute values | e* | will be smaller than a given positive number e, provided the distance between two consecutive points of division of the curve is less than the corresponding number 17, and the formula (2) will be established II, 27] DEFINITE INTEGRALS 65 where /(), <j>(t), $(t) are three real functions of the real variable t continuous in the interval (a, ft). From the veiy definition of the integral we evidently have 1 = f /(*) < 09 dt + * f */<r ,Ar Let us suppose, for defimteness, that a < ft , then a is the length of the path of integration measured from a, and the general formula which gives an upper bound for the absolute value of a definite integral becomes or, supposing that/() is positive between a and \i\s Applying the law of the mean to this new integral, and indicating by f a value of t lying between a and ft, we have also m^ Setting F(t) = 4>() -f- i\lr(), this result may also be written in the form (3) I~\F(f) ("ffidt, Ja where X is a complex number whose absolute value is less than or equal to unity, this is Darboux's formula. To Weierstrass is due a more precise expression, which has a rela- tion to some elementary facts of statics When t varies from a to ft, the point with the coordinates x = <f> (), y = \fr (t) describes a certain curve L Let (X Q , y ), (x v y^ - , (x t _ 19 y^-i); - be the points of L which correspond to the values a, t v , ^_ 1? - - of t, and let According to a kaown theorem, Z" and F are the coordinates of the center of gravity of a system of masses placed at the points (# , y ), (x v yj), , (#*_!, ^-0, of the curve X, the mass placed at the point (aj^j, y^j) being equal to f(t k ^(t k - t^J, where /(*) is 66 THE GENERAL CAUGHT THEOBY [n,27 still supposed to be positive It is clear that the center of gravity lies within every closed convex curve C that envelops the curve L When the number of mteivals increases indefinitely, the point (X, Y) will have for its limit a point whose coordinates (u, v) are given by the equations ~ JLVO* which is itself within the curve (7. We can state these two formulae as one by wilting / n& (4) I = (u 4- w) / /OO dt = Z I /(*) dt, Ja <J wheie ^ is a point of the complex plane situated with in every closed convex cuwe enveloping the curve L It is clear that, in the general case, the factoi Z of Weierstrass is limited to a much more lestricted region than the factor AJF() of Darboux 28. Integrals taken along a closed curve. In the preceding para- graphs, it suffices to suppose that /(#) is a continuous function of the complex variable # along the path of integration We shall now suppose also that /() is an analytic function, and we shall first con- sidei how the value of the definite integral is affected by the path followed by the vanable in going from A to E If a function f(z) Is analytic within a dosed curve and also on the curve itself y th& integral Jf{&)d& y taken around that curve } w egrual to zero In order to demonstrate this fundamental theorem, which is due to Canchy, we shall first establish several lemmas : 1) The integrals fdz, Jz dz, taken along any closed curve what- ever, are zero In fact, by definition, the integral fda, taken along any path whatever between the two points a, b, is equal to b a, and the integral is zero if the path is closed, since then I = a. As for the integral /# d&, taken along any curve whatever joining two points a, I, if we take successively f ft = s^_ l9 then 4 = z k ( 25), we see that the integral is also the limit of the sum -n(X.f i - O __ yaf+i - ^ _ & - a* ~ hence it is equal to zero if the curve is closed. 2) If the region bounded by any curve C whatever be divided into smaller parts by transversal curves drawn arbitianly, the sum of tlie integrals ff(z)dz taken in the same sense along the boundary II, 28] DEFINITE INTEGRALS 67 of each of these parts is equal to the integral //() dz taken along the complete boundaiy C. It is clear that each portion of the auxil- iary curves sepaiates two contiguous regions and must be described twice in integration in opposite senses. Adding all these inte- grals, there will remain then only the integrals taken along the boundary curve, whose sum is the integial f^f()dz Let us now suppose that the region A is divided up, partly in smaller regular paits, which shall be squares having their sides parallel to the axes Ox, Oy , partly in irregular parts, which shall be portions of squares of which the remaining part lies beyond the boundaiy C. These squares need not necessarily be equal For ex- ample, we might suppose that two sets of parallels to Ox and Oy have been drawn, the distance between two neighboimg parallels being constant and equal to Z , then some of the squares thus obtained might be divided up into smaller squares by new parallels to the axes Whatever may be the manner of subdivision adopted, let us suppose that there are N regular parts and N 1 irregular parts , let us number the regular parts in any order whatever from 1 to A r , and the irregular parts from 1 to A T ' Let Z t be the length of the side of the zth square and Z that of the square to which the A,th irregular part belongs, L the length of the boundary C, and Jl the area of a polygon which contains within it the curve C. Let abed be the ^th square (Fig. 13), let # t be a point taken in its interior or on one of its sides, and let 2 be any point on its boundary, Then we have FIG. 13 where |e,| is small, provided that the side of the square is itself small. It follows that /() = */(*,) +/(*,) - *,/'(*,) +,(*- *,), 68 THE GENERAL CATJCHY THEORY [II, 28 =/(*) f /<c t ) where the integrals are to be taken along the perimetei c t of the square By the first lemma stated above, this reduces to the form (6) Again, let pqrst be the &th irregular part, let z[ be a point taken in its interior or on its perimeter, and let z be any point of its perimeter Then we have, as above, where e* is infinitesimal at the same tune as l' k ; whence we find (8) Let t] be a positive number greater than the absolute values of all the factors c t and e* The absolute value of z s t is less than Z l V2 , hence, by (6), we find where <u t denotes the area of the tth regular part. From (8) we find, in the same way, : tfi V2 (4 Zfc + arcrs) = 4 77 VI <* + ^ V2 arcrc, where c^ is the area of the square which contains the kfh irregular part. Adding all these integrals, we obtain, a fortiori, the inequality < t) [4 V2 (Su, + 20 + X V2Z], where X is an upper bound for the sides l' k When the number of squares is increased indefinitely in such a way that all the sides Z t and l' k approach zero, the sum So> t + 54 finally becomes less than Jl On the right-hand side of the inequality (9) we have, then, the product of a factor which remains finite and another factor 77 which can be supposed smaller than any given positive number This can be true only if the left-hand side is zero ; we have then II, 29] DEFINITE INTEGRALS 69 29. In order that the preceding conclusion may be legitimate, we must make sure that we can take the squares so small that the absolute values of all the quantities c t , e \v ill be less than a positive number 17 given in advance, if the points Zi and z' k are suitably chosen.* We shall say for brevity that a region bounded by a closed curve 7, situated in a region of the plane inclosed by the curve C, satisfies the condition (a) with respect to the number 17 if it is possible to find in the interior of the curve 7 or on the curve itself a point asf such that we always have (ex) !/(*) when z describes the curve 7 The proof depends on showing that we can choose the squares so small that all the parts considered, regular and irregular, satisfy the condition (a) with respect to the number ij. We shall establish this new lemma by the well-known process of successive subdivisions Suppose that we have first drawn two sets of parallels to the axes Ox, Oy, the distance between two adjacent parallels being constant and equal to I Of the parts obtained, some may satisfy the condition (a), while others do not. Without changing the parts which do satisfy the condition (cr), we shall divide the others into smaller parts by joining the middle points of the opposite sides of the squares which form these parts or which inclose them If, after this new operation, there are still parts which do not satisfy the condition (a), we will repeat the operation on those parts, and so on Continuing in this way, there can be only two cases either we shall end by having only regions which satisfy the condition (a), in which case the lemma is proved ; or, however far we go in the succession of operations, we shall always find some parts which do not satisfy that condition. In the latter case, in at least one of the regular or irregular parts obtained by the first division, the process of subdivision ]ust described never leads us to a set of regions all of which satisfy the condition (a) ; let A 1 be such a part After the second subdivision, the part A l contains at least one subdivision -4 2 which cannot be subdivided into regions all of which satisfy the condition (a) Since it is possible to continue this reasoning indefinitely, we shall have a suc- cession of regions ^ii "^2? -^-si * "^ n * " which are squares, or portions of squares, such that each is included in the pre- ceding, and whose dimensions approach zero as n becomes infinite. There is, therefore, a limit point z situated in the interior of the curve or on the curve itself Since, by hypothesis, the f unction /(z) possesses a derivative f(z Q ) for z = # , we can find a number p such that provided that | z z 1 is less than p Let c be the circle with radius p described about the point z as center For large enough values of TI, the region A n will lie within the circle c, and we shall have for all the points of the boundary of A n \fto-f to- <?-**) fW>l^\*-*9\* * GOUBSAT, Transactions of the American Mathematical Society t 1900, Vol I, p 14. 70 THE GENERAL CAUGHT THEORY [II, 29 Moreover, it is clear that the point Z Q is in the interior of A n or on the boundary , hence that region must satisfy the condition (a) with respect to 17 We are therefore led to a contradiction in supposing that the lemma is not true 30. By means of a suitable convention as to the sense of integra- tion the theorem can be extended also to boundaries formed by several distinct closed curves Let us consider, for example, a func- tion /(*) analytic within the region A bounded by the closed curve C and the two interior curves C", C", and on these curves themselves (Fig 14). The complete boundary T of the region A is formed by these three distinct curves, and we shall say that that boundary is described in the positive sense if the legion A is on the left hand with respect to this sense of motion ; the arrows on the figure indicate the positive sense of description for each of the curves. With this agree- ment, we have always f ^ (r) - 0, p IG 14 the integral being taken along the complete boundary in the positive sense. The proof given for a region with a simple boundary can be applied again here , we can also reduce this case to the preceding by drawing the transversals al, cd and by applying the theorem to the closed curve abmbandcpcdqa (I, 153). It is sometimes convenient m the applications to write the preced- ing formula in the form = f /(*)&+ r J<C"> Jcc where the three integrals are now taken in the same sense ; that is, the last two must be taken in the reverse direction to that indicated by the arrows. Let us return to the question proposed at the beginning of 28 ; the answer is now very easy Let f(z) be an analytic function in a region j4 of the plane. Given two paths AMB 9 ANB, having the same extremities and lying entirely in that region, they will give the same value for the integral ff(z)d& if the function /() is analytic within the closed curve formed by the path AMB followed by the path BNA. We shall suppose, for defimteness, that that closed curve II, 30] DEFINITE INTEGRALS 71 does not have any double points Indeed, since the sum of the two integrals along ylAfjS and along BXA is zero, the two integials along AMB and along ANB must be equal. We can state this lesult again as follows Two paths AMB and ANB, having the same extremities, give the same value for the integral ff(z)d& if we can pass from one to the other by a continuous deformation without encountering any point where the function ceases to b& analytic This statement holds true even when the two paths have any num- ber whatever of common points besides the two extremities (I, 152). Fiom this we conclude that, when /(?) is analytic in a region bounded by a single closed curve, the integral ff(?)dz is equal to zero when taken along any closed curve whatever situated in that region But we must not apply this result to the case of a region bounded by several distinct closed curves. Let us consider, for exam- ple, a function f(z) analytic in the ring-shaped region between two concentric circles C, C*. Let C" be a circle having the same center and lying between C and C r ; the integral ff(z) dz 9 taken along C", is not in geneial zeio. Cauchy's theoiem shows only that the value of that integral remains the same when the radius of the cucle C" is varied.* * Cauehy's theorem remains true without any hypothesis upon the existence of the function/ (z) beyond the legion A limited by the cuive C, 01 upon the existence of a derivative at each point of the curve C itself It is sufficient that the function/ (z) shall be analytic at every point of the region J., and continuous on the boundary C, that is, that the value /(Z) of the function in a point Z of C varies continuously with the position of Z on that boundary, and that the difference/ (Z) -/(), where z is an interior point, approaches zero uniformly with \Z z\ In fact, let us first suppose that every straight line from a fixed point a of A meets the boundary in a single point When the point z describes C, the point a + 6 (z- a) (where is a real number between and 1) describes a closed curve C' situated in A The difference between the two integrals, along the curves C and C7', is equal to and we can take the difference 1-0 so small that \S] will be less than any given positive number, for we can write the function under the integral sign in the foim Since the integral along (7 is zero, we have, then, also f ^C In the case of a boundary of any form whatever, we can leplace this boundary by a succession of closed curves that fulfill the preceding condition by drawing suitably placed transversals 72 THE GENERAL CAUCHY THEORY [II, 31 31. Generalization of the formulae of the integral calculus. Let/() be an analytic function in the region A limited by a simple boundary curve C. The definite integral taken from a fixed point * up to a variable point ^ along a path lying in the region A 9 is, from what we have just seen, a definite function of the upper limit Z We shall now show that this function *(2T) is also an analytic function of Z whose derivative is f(Z) For let ^ + h be a point near - , then we have and we may suppose that this last integral is taken along the seg- ment of a straight line joining the two points Z and Z + h. If the two points are very close together, / (s) differs very little from/(Z) along that path, and we can write /(*)=/(*) + 8, where I SI is less than any given positive number ^ provided that \h\ is small enough Hence we have, after dividing by A, The absolute value of the last integral is less than iy|&|, and there- fore the lefkhand side has for its limit f(Z} when 7i approaches zero If a function F(Z) whose derivative is/(Z) is already known, the two functions *(2) and P(Z) differ only by a constant (footnote, p. 38), and we see that the fundamental formula of integral calculus can be extended to the case of complex variables (10) A*) ** = *(*i) - F (*o) J This formula, established by supposing that the two functions f(z), F(z) were analytic in the region A, is applicable in more general cases. It may happen that the function JP(), or both/(s) and P() at the same time, are multiple-valued ; the integral has a precise meaning if the path of integration does not pass through any of the critical points of these functions. In the application of the formula it will be necessary to pick out an initial determination ^( ) of the primitive function, and to follow the continuous variation of that II, 31] DEFINITE INTEGRALS 73 function when the variable describes the path of integration. Moreover, if f(z) is itself a multiple-valued function, it will be neces- sary to choose, among the determinations of F(z), that one whose derivative is equal to the determination chosen for/(z). Whenever the path of integration can be inclosed within a region with a simple boundary, in which the branches of the two functions f(z), jF(z) under consideration are analytic, the formula may be regarded as demonstrated Now in any case, whatever may be the path of integration, we can break it up into several pieces for which the preceding condition is satisfied, and apply the formula (10) to each of them separately Adding the results, we see that the for- mula is true in general, provided that we apply it with the necessary precautions. Let us, for example, calculate the definite integral *&*&, taken along any path whatever not passing through the origin, where t m is a real or a complex number different from 1 One primitive func- tion is 3 m+1 /(ra + 1), and the general formula (10) gives In order to remove the ambiguity present in this formula when m is not an integer, let us write it in the form Wz = - The initial value Log( ) having been chosen, the value of & m is thereby fixed along the whole path of integration, as is also the final value Log^). The value of the integral depends both upon the initial value chosen for Log ( ) and upon the path of integration. Similarly, the formula <b = Log [/(*,)] - Log [/(* )] does not present any difficulty in interpretation if the function f(z) is continuous and does not vanish along the path of integration The point u =/() describes in its plane an arc of a curve not pass- ing through the origin, and the right-hand side is equal to the vari- ation of Log(w) along this arc Finally, we may remark in passing that the formula for integration by parts, since it is a consequence of the formula (10), can be extended to integrals of functions of a complex variable 74 THE GENERAL CAUCHr THEORY [II, 32 32. Another proof of the preceding results. The properties of the integral //()rf* present a gieat analogy to the pioperties of line integrals when the condition for mtegrability is fulfilled (I, 152). Eiemann has shown, in fact, that Cauchy's theorem results im- mediately from the analogous theorem relative to line integrals. Let /() = J + Yi be an analytic function of z within a region A with a simple boundary , the integral taken along a closed curve C lying in that region is the sum of two line integrals . /(*)&= f Xcbs-Ydy + i C Ydx + Xdy, > J(O J(C) and, from the relations which connect the denvatives of the func- tions X, Y, ^ __ 1 <?lE __ ? dx dy dy dx we see that both of these line integrals are zero * (I, 152) It follows that the integral f*f(z)dz, taken from a fixed point # to a variable point , is a single-valued function <(V) in the region A Let us separate the real pait and the coefficient of i in that function . /(*, y) /tey) P(x, y)s= / Xdx - Ydij, Q(x, y;= I Ydx+Xdy. A*o'0o> *A a o'y<r> The functions P and Q have partial derivatives, J!_ Y ! __ Q_ Q_ ^r jtL, "T^ J., JC, o^" -^-t ^a; oy ' ox oy ' which satisfy the conditions ap_aa ^__^ 3i ^y Sy "" fix Consequently, P + Qi is an analytic function of & whose derivative If the function /(#) is discontinuous at a certain number of points of A, the same thing will be true of one or more of the functions X l 7", and the line integrals P(x, y), Q(x, y) will in general have periods that arise from loops described about points of discontinuity (I, 153) The same thing will then be true of the integral f x z f(z) dz We shall resume the study of these periods, after having investigated the nature of the singular points of /(). * It should be noted that Biemann's proof assumes the continuity of the deriva- , dY/dy, , that is, of/'(z) It 33] THE CAUCHY INTEGRAL THEOREMS 75 To give at least one example of this, let us consider the integral f^dz/z. After separating the real part and the coefficient of t, we have ycte r z dz_ __ r (* y)(c + idy __ r <* v>xdx+ ydy Ji z J(i, o) x + ly ~~ J(i, o) a 2 + y 2 (i, o) The real part is equal to [log (a 2 4- y 2 )]/2, whatever may be the path followed. As for the coefficient of i, we have seen that it has the period 2 ir , it is equal to the angle through which the radius vector joining the origin to the point (x, y) has turned We thus find again the various determinations of Log(z). II CAUCHY'S INTEGRAL TAYLOR'S AND LAURENT'S SERIES SINGULAR POINTS RESIDUES We shall now present a series of new and important results, which Cauchy deduced from the consideration of definite integrals taken between imaginary limits. 33. The fundamental formula. Let/(s) be an analytic function in the finite region A limited by a boundary r, composed of one or of several distinct closed curves, and continuous on the boundary itself If a; is a point * of the region A, the function is analytic in the same region, except at the point z = x With the point x as center, let us describe a circle y with the radius p, lying entirely in the legion A , the preceding function is then analytic in the region of the plane limited by the boundary r and the circle y, and we can apply to it the geneial theorem ( 28). Suppose, for defmiteness, that the boundary r is composed of two closed curves (7, ' (Fig. 15) Then we have /<*)** ! r /(*) *-* JM Z - where the three integrals are taken in the sense indicated by the arrows. We can write this in the form *)fo = r /(*) -* ./&>*- * In what follows we shall often have to consider several complex quantities at the same time We shall denote them indifferently by the letters , z, u, Unless it is expressly stated, the letter x will no longer be reserved to denote a real variable. 76 THE GENERAL CAUCHY THEORY [II, 33 where the integral L. denotes the integral taken along the total boundary T in the positive sense If the radius p of the circle y is very small, the value of f(&) at any point of this circle differs very hfctle from /(). /()=/(*) + 8, where |8| is very small. Replacing /(s) by this value, we find The first integral of the right-hand side is easily evaluated , if we put z = x 4- pe ei j it becomes *" . == 2 TTl The second integral J^S dz/(z x) is therefore independent of the radius p of the circle y, on the other hand, if |S| remains less than 15 a positive number 17, the absolute value of this integral is less than (y/p) 2 Trp = 2 7Ti7 Now, since the function f(z) is continuous for z = x, we can choose the radius p so small that 77 also will be as small as we wish Hence this integral must be zero Dividing the two sides of the equation (11) by 2 TTI, we obtain (12) This is Cauchy's fundamental formula It expresses the value of the function /(*) at any point x whatever within the boundary by means of the values of the same function taken only along that boundary Let x + Ace be a point near x, which, for example, we shall suppose lies in the interior of the circle y of radius p Then we have also II, 33] THE CAUCHY INTEGRAL THEOREMS 77 and consequently, subtracting the sides of (12) from the correspond- ing sides of this equation and dividing by Ace, we find /(s)_ 1 r f(z)dz Ax 2 m J (r) (z x) (z x Aa) " When Ace approaches zero, the function under the integral sign ap- proaches the limit f(z)/(z xf In order to prove rigorously that we have the right to apply the usual formula for differentiation, let us write the integral in the form *)<&* , 2 Let M be an upper bound for \f(z)\ along r, L the length of the boundary, and 8 a lower bound for the distance of any point what- ever of the circle y to any point whatever of r The absolute value of the last integral is less than ML\&x\/$* and consequently ap- proaches zero with |Aa?|. Passing to the limit, we obtain the result ^QN (13) It may be shown in the same way that the usual method of differ- entiation under the integral sign can be applied to this new integral * and to all those which can be deduced from it, and we obtain successively and, in general, & X)" + 1 Hence, if a function /() is analytic in a certain region of the plane, the sequence of successive derivatives of that function is unlimited, and all these derivatives are also analytic functions in the same region It is to be noticed that we have arrived at this result by assuming only the existence of the first derivative. Note. The reasoning of this paragraph leads to more general con- clusions Let <() be a continuous function (but not necessarily * The general formula for differentiation under the integral sign will be established later (Chapter V) 78 THE GENERAL CAUCHY THEORY [II, 33 analytic) of the complex variable 2 along the curve T, closed or not. The integral has a definite value for every value of x that does not lie on the path of integration. The evaluations just made prove that the limit of the quotient [F(x + Aas) JP(a?)]/Aoj is the definite integral when |Aoj| approaches zero Hence F(%) is an analytic function for every value of x, except for the points of the curve T, which are in general singular points for that function (see 90) Similarly, we find that the nth derivative F^(x) has for its value 34. Mbrera's theorem. A converse of Cauchy's fundamental theorem which was first proved "by Morera may be stated as follows If a Junction f(z) of a complex variable z is continuous in a region A, and if the definite integral f^f(z) dz, taken along any closed curve G lying in -4., is zero, then f(z) is an analytic Junc- tion in A . For the definite integral F(z) = J]/(*)cK, taken between the two points , z of the region A along any path whatever lying in that region, has a definite value independent of the path If the point Z Q is supposed fixed, the integral is a function of z The reasoning of 31 shows that the quotient AF/Az has f(z) for its limit when Az approaches zero Hence the function F(z) is an analytic function of z having f(z) for its derivative, and that derivative is therefore also an analytic function 35. Taylor's series. Let f(z) be an analytic function in the interior of a circle with the center a , the value of that functwn at any point x within the circle is equal to the sum of the convergent series In the demonstration we can suppose that the function f(z) is analytic on the circumference of the circle itself ; in fact, if x is any point in the interior of the circle C, we can always find a circle C' 9 with center a and with a radius less than that of (7, which contains II, 35] THE CAUCHY-TAYLOB, SERIES 79 the point x within it, and we would reason with the circle C 1 just as we are about to do with the circle C. With this undei standing, x being an interior point of C, we have, by the fundamental formula, Let us now write !/( x) in the following way or, carrying out the division up to the remamdei of degree n -f- 1 in x a, 1 _ 1 \ x ~ a i OP a) 3 i ^r^-s-a + ^-^ + ^-a) 8 " 1 " + (a -a)* { (x-aY^ Let us replace l/(* a?) in the formula (12^ by this expression, and let us bring the factors x a, (x a) 3 , * , independent of z, outside of the integral sign. This gives where the coefficients J" , / 1? - , / and the remainder R n have the values (16) ^ n As n becomes infinite the remainder R n approaches zero For let M be an upper bound for the absolute value of /(*) along the circle (7, R the radius of that circle, and r the absolute value of x a. We have | * - a? | ^R - r, and therefore 1 1/(* - x) \ ^1/(R - r), when * describes the circle C. Hence the absolute value of R n is less than 1 M _ MR /7-Y+ 1 and the factor (r/R) n + l approaches zero as n becomes infinite. From this it follows that f(x) is equal to the convergent series 80 THE GENERAL CAUCHY THEORY [II, 35 Now, if we put x = a in the formulae (12), (13), (14), the boundary T being here the circle C, we find The series obtained is therefore identical with the series (15) ; that is, with Taylor's series The circle C is a circle with center a, in the interior of which the function is analytic, it is clear that we would obtain the gieatest circle satisfying that condition by taking for radius the distance from the point a to that singular point of f(z) nearest a This is also the circle of convergence for the series on the right * This important theorem brings out the identity of the two defini- tions for analytic functions which we have given (I, 197, 2d ed , 191, 1st ed , and II, 3) In fact, every power series represents an analytic function inside of its circle of convergence ( 8) , and, conversely, as we have ]ust seen, every function analytic in a circle with the center a can be developed in a power series proceeding according to powers of x a and convergent inside of that circle Let us also notice that a certain number of results previously estab- lished become now almost intuitive; for example, applying the theorem to the functions Log (1 + &) and (1 + #) m , which are ana- lytic inside of the circle of unit radius with the origin as center, we find again the formulae of 17 and 18 Let us now consider the quotient of two power series /(#)/< (x), each convergent in a circle of radius R. If the series <j>{x) does not vanish for x = 0, since it is continuous we can describe a circle of radius r 35 R in the whole interior of which it does not vanish The f unction /(#)/< (x) is therefore analytic in this circle of radius r and can therefore be developed in a power series in the neighborhood of the origin (I, 188, 2d ed ; 183, 1st ed ) In the same way, the theorem relative to the substitution of one series in another series can be proved, etc Note Let/(#) be an analytic function in the interior of a circle C with the center a and the radius r and continuous on the circle itself. The absolute value | f(&) | of the function on the circle is a continuous function, the maximum value of which we shall indicate by M(r). On the other hand, the coefficient a n of (x a) n in the * This last conclusion requires some explanation on the nature of singular points, which will be given in the chapter devoted to analytic extension II, 37] THE CAUCHY-LAURENT SERIES 81 development of /() is equal to /<"> ()/', that is, to we have, then, (17) A n = \a n \< so that JXC(r) is greater than all the products A n r** We could use 3fC(r) instead of M in the expression for the dominant function (I, 186, 2d ed , 181, 1st ed ) 36 Liouville's theorem If the function f(x) is analytic for every finite value of x } then Taylor's expansion is valid, whatever a may be, in the whole extent of the plane, and the function considered is called an ^ntegral function From the expressions obtained for the coeffi- cients we easily derive the following proposition, due to Liouville : Every integral function whose absolute, value is always less than a fixed number M is a constant. For let us develop f(x) in powers of x a, and let a n be the coefficient of (x a) n It is clear that <%C(r) is less than M, what- ever may be the radius r, and therefore \a n \ is less than Jlf/r*. But the radius r can be taken just as large as we wish , we have, then, a n = if n ^ 1, and f(x) reduces to a constant /(a). More generally, let f(x) be an integral function such that the absolute value of f(x)/x m remains less than a fixed number M for values of x whose absolute value is greater than a positive number R ; then thefunctwnf(x) is a polynomial of degree not greater than m. For suppose we develop f(x) in powers of x, and let a n be the coefficient of x n . If the radius r of the circle C is greater than J?, we have JXC(r) < Mr, and consequently |<z n | < Mr-" If n > m, we have then a n = 0, since Mr"* can be made smaller than any given number by choosing r large enough 37. Laurent's series. The reasoning by which Cauchy derived Taylor's series is capable of extended generalizations. Thus, let f(z) be an analytic function in the ring-shaped region between the * The inequalities (17) are interesting, especially since they establish a relation between the order of magnitude of the coefficients of a power series and the order of magnitude of the function, 5W(r) is not, in general, however, the smallest number which satisfies these inequalities, as is seen at once when all the coefficients a n are real and positive These inequalities (17) can be established without making use of Cauchy 's integral (MERAY, Legons nouvelles sur I' analyse irtfimtisfimale, Vol I, p 99). 82 THE GENERAL CAUCHY THEORY [n, 37 two concentric circles C, C' having the common center a We shall show that the value f(x) of the function at any point x taken in that region is equal to the sum of two convergent series, one proceeding in positive powers ofx a, the other in positive powers ofl/(x a) * We can suppose, just as before, that the function /(#) is analytic on the circles f, C' themselves Let E, R 1 be the radii of these circles and r the absolute value of x a , if C 1 is the interior circle, we have jR' < r < R About x as center let us describe a small circle -y lying entirely between C and C' We have the equality the integrals being taken in a suitable sense , the last integral, taken along y, is equal to 2 7rif(x) } and we can write the preceding relation m the form (18) /(*)' where the integrals are all taken in the same sense Repeating the reasoning of 35, we find again that we have where the coefficients J 03 J" i; -, J n , are given by the formulae (16) In order to develop the second integral in a series, let us notice that x and that the integral of the complementary term, i r (i^ym^ * m JwV* a ' x ~~* approaches zero when n increases indefinitely In fact, if M r is the maximum of the absolute value of /(#) along C", the absolute value of this integral is less than * Comptes rendus de I'Academw des Sciences, Vol XVII See OEuvres de Cauehy, 1st senes, Vol VHI, p 115 II, 37] THE CAUCHY-LAURENT SERIES 83 and the factor R'/r is less than unity. We have, then, also (20 ) JL ^ > where the coefficient K n is equal to the definite integral Adding the two developments (19) and (20), we obtain the proposed development of f(x) In the formulae (16) and (21), which give the coefficients / and K n , we can take the integrals along any cucle r whatever lying between C and C f and having the point a for center, for the functions under the integral sign are analytic in the ring. Hence, if we agree to let the index n vary from oo to-fao, we can write the development of f(x) in the form (22) /(*)= + " i where the coefficient / w , whatever the sign of n, is given by the formula Example The same f unction /(x) can have developments which are entirely different, according to the region considered Let us take, for example, a rational fraction /(a;), of which the denominator has only simple roots with different absolute values. Let a, &, c, , I toe these roots arranged in the order of increasing absolute values Disregarding the integral part, which does not interest us here, we have + JL + x a a? 6 oj c a I In the circle of radius a about the origin as center, each of the simple frac- tions can be developed in positive powers of x, and the development off(x) is identical with that given by Maclaunn's expansion In the ring between the two circles of radii | a\ and |6| the fractions l/(x 6), l/(x c), - , l/(x Z) can be developed m positive powers of x, but l/(x a) must be developed in positive powers of 1/x, and we have 84 THE GENERAL CAUCHY THEORY [II, 37 In the next ring we shall have an analogous development, and so on Finally, exterior to the circle of radius |/|, we shall have only positive powers of 1/x /(*) = + L Aa,+ 38. Other series. The proofs of Taylor's series and of Laurent's series are based essentially on a particular development of the simple fraction l/(z x) when the point x remains inside or outside a fixed circle Appell has shown that we can again generalize these formulae by considering a function f(x) analytic in the inteiior of a region A bounded by any number whatever of arcs of FIG 16 circles or of entire circumferences * Let us consider, for example, a function /(x) analytic in the curvilinear triangle PQR (Fig 16) formed by the three arcs of circles PQ, QB, JBP, belonging respectively to the three circumferences (7, C", C" Denoting by % any point within this curvilinear triangle, we have - 1 C f(z)ds + 1 C /(z)(fe i * C f(e)d z) -^ / TTF+R; L, .-_._i L, T= Along the arc PQ we can write 1 1 x 2 a e-a ~r ~7~ To T ' a) 2 1 /x a' >, x \z a, where a is the center of C , but when 2 describes the arc PQ, the absolute value of (x a)/(z a) is less than unity, and therefore the absolute value of the integral approaches zero as n becomes infinite. We have, therefore, Vol I, p 145 II, 38] THE CAUCHY-LAUREtfT SERIES 85 where the coefficients are constants whose expressions it would be easy to write out Similarly, along the arc QR we can write x-z x- ~ /-. t\^ ' I /z-b\ X Z \X &/ where 6 is the center of C" Since the absolute value of (z b) n /(x 6) ap- proaches zero as n becomes infinite, we can deduce from the preceding equation a development for the second integral of tie form /m J_ C f&te *! r t Jr. Similarly, we find j_ 2 * where c is the center of the circle C". Adding the three expressions (a), (), (7), we obtain foi/(x) the sum of three semes, proceeding respectively accord- ing to positive powers of x - a, of l/(x - 6), and of l/(x c). It is clear that we can transform this sum into a series of \rtuch all the terms are rational func- tions of a, for example, by uniting all tke terms of the same degree in x a, l/(z &), l/(x c) The preceding reasoning applies whatever may be the number of arcs of circles It is seen in the preceding example that the three series, (a), (0), (7), are still convergent when the point x is inside the triangle P'Q'JB', and the sum of these three series is again equal to the integral taken along the boundary of the triangle PQR m the positive sense. Now, when the point x is in the triangle f^Q'J?', th& function f(z)/(z x) is analytic in the interior of the triangle PQJS, and the preceding integral is therefore zero Hence we obtain in this way a series of ra,tional fractions which is convergent when x is within one of the two triangles JPQR, P'Q'.R', and for which the sum i& equal tof(x) or to zero, according as thepwnt xisin the triangle PQE or in tTie triangle P'QR' Pamleve* has obtained more general results along the same lines * Let us con- sider, m order to limit ourselves to a very simple case, a convex closed curve T having a tangent which changes continuously and a radius of curvature which remains under a certain upper bound It is easy to see that we can associate with each point M of r a circle C tangent to T at that point and inclosing that curve entirely in its interior, and this may be done in such a way that the center of the circle moves in a continuous manner with M Let/(z) be a function ana- lytic in the interior of the boundary T and continuous on the boundary itself. Then, m the fundamental formula * Sur les hgnes slngulures desfonctiow malytiques (Annates de laFacultf de Toulouse, 1888) 86 THE GENERAL CAUGHT THEORY [II, 38 where x is an interior point to r, we can write I 1 x-a (x-a) n 1 /x-a\+i .+ - +. . + /- :'+()" z x z a (z a) 2 (z a) n + - 1 z a? \z a/ where a denotes the center of the circle C which corresponds to the point z of the boundary , a is no longer constant, as in the case already examined, but it is a continuous function of z when the point M describes the curve T - Never- theless, the absolute value of (x a)/(z a), which is a continuous function of z, remains less than a fixed number p less than unity, since it cannot reach the value unity, and therefore the integral of the last term approaches zero as n becomes infinite Hence we have (25) and it is clear that the general term of this series is a polynomial P n (x) of degree not greater than n The function f(x) is then developable in a series of polynomials in the interior of the boundary T The theory of conformal transformations enables us to obtain another kind of series for the development of analytic functions Let f(x) be an analytic function in the interior of the region J., which may extend to infinity Suppose that we know how to represent the region A confoimally on the region inclosed by a circle C such that to a point of the region A corresponds one and only one point of the circle, and conversely , let u = <f> (z) be the analytic function which establishes a correspondence between the region A and the circle C hav- ing the point u = for center m the w-plane When the variable u describes this circle, the corresponding value of z is an analytic function of u The same is true of /(), which can therefore ( be developed in a convergent series of powers of u, or of (2), when the variable z remains in the intenor of A Suppose, for example, that the region A consists of the infinite strip included between the two parallels to the axis of reals y = 0. We have seen ( 22) that by putting u = (#*>* !)/(&*&* + 1) this strip is made to correspond to a circle of unit radius having its center at the point u = Every function analytic in this strip can therefore be developed in this strip in a convergent series of the following form 39. Series of analytic functions. The sum of a uniformly conver- gent series whose terms are analytic functions of is a continuous function of z, but we could not say without further proof that that sum is also an analytic function It must be proved that the sum has a unique derivative at every point, and this is easy to do by means of Cauchy's integral. Let us first notice that a uniformly convergent series whose terms are continuous functions of a complex variable # can be integrated term by term, as in the case of a real variable The proof given in II, 39] THE CAUCHY-LAURENT SERIES 87 the case of the real variable (I, 114, 2d ed ; 174, 1st ed ) applies here without change, provided the path of integration has a finite length The theorem which we wish to prove is evidently included in the following more general proposition Let be a series all of whose terms are analytic functions in a region A bounded by a closed curve F and continuous on the boundary. If the series (26) is uniformly convergent on T, it is convergent in every point of A, and its sum is an analytic function F(&) whose jt?th derivative is represented by the series fonned by the ^>th derivatives of the terms of the series (26). Let <j> (2) be the sum of (26) in a point of r , < (#) is a continuous function of & along the boundary, and we have seen ( 33, Note) that the definite integral (2T) , (s) ^ J ^ } where x is any point of A, represents an analytic function in the region A, whose pfh. derivative is the expression (2& (28) Since the series (26) is uniformly convergent on T, the same thing is true of the series obtained by dividing each of its terms by z x, and we can write or again, since f v () is an analytic function in the interior of r, we have, by formula (12), Similarly, the expression (28) can be written in the form Hence, if the series (26) is uniformly convergent m a region A of the plane, x being any point of that region, it suffices to apply the 88 THE GENERAL CAUCHY THEORY [II, 39 preceding theorem to a closed curve T lying in A and suuoundmg the point x. This leads to the following pioposition Every series uniformly convergent in a region A of the plane, whose terms are all analytic functions in A, represents an analytic function F(z) in the same region The pfh derivative of F(e) is equal to the series obtained by differentiating p times each tenn of the series which represents F(z) * 40. Poles. Every function analytic in a circle with the center a is equal, in the interior of that circle, to the sum of a power series (29) /(*)=^ + ^(*-a) + - +^.(*-a)" + We shall say, for brevity, that the function is regular at the point a, or that a is an ordinary point for the given function We shall call the interior of a circle 0, descubed about a as a center with the radius p, the neighborhood of the point a, when the formula (29) is applicable. It is, moreover, not necessary that this shall be the largest circle in the interior of which the formula (29) is true , the radius p of the neigh- borhood will often be defined by some other particular property If the first coefficient A Q is zero, we have f(a) = 0, and the point a is a zero of the function /() The order of a zero is defined in the same way as for polynomials ; if the development of f(z) commences with a term of degree m in a } /(*) = ^(*-) + ^ + i(*-)" + 1 + ., (m > 0), where A m is not zero, we have /(a) = 0, /'() = 0, ., /*-() =0, y*->(a)*0, and the point a is said to be a zero of order m We can also write the preceding formula in the form < (&) being a power series which does not vanish when = a Since this series is a continuous function of z, we can choose the radius p of the neighborhood so small that <(#) does not vanish m that neighborhood, and we see that the function /() will not have any other zero than the point a in the interior of that neighborhood. The zeros of an analytic function are therefore isolated points Every point which is not an ordinary point for a single-valued f unction f(z) is said to be a singular point A singular point a of the * This proposition is usually attributed to Weierstrass II, 40] SINGULAR POINTS 89 f unction /(g) is &pole if that point is an ordinary point for the re- ciprocal function !//(). The development of !//() in poweis of a cannot contain a constant term, for the point a would then be an ordinary point for the function f(z) Let us suppose that the development commences with a term of degree m in z a, (30) 7^ = (-)-*(*), where <() denotes a regular function in the neighboihood of the point a which is not zero when z = a. From this we derive (31) /(*) = where \ff(z) denotes a regular function in the neighboihood of the point a which is not zero when z = a. This formula can be written in the equivalent form (SI 1 ) /(*)=, A \ OT + / Bm \l 1 + - x ' ' v y m - 1 a where we denote by P(z a), as we shall often do hereafter, a regular function for z = a, and by J3 m , B m _ 19 -, ^ ceitain con- stants Some of the coefficients B 19 B^ - , B m _ 1 may be zero, but the coefficient B m is surely different from zero. The integer m is called the order of the pole It is seen that a pole of order m of f(z) is a zero of order m of !//"(), and conversely. In the neighborhood of a pole a the development of f(z) is com- posed of a regular part P(z a) and of a polynomial in l/(s a)j this polynomial is called the principal part of /(#) in the neighbor- hood of the pole. When the absolute value of a approaches zero, the absolute value off(z) becomes infinite in whatever way ike point approaches the pole In fact, since the function ij/(z) is not zero for & = a, suppose the radius of the neighborhood so small that the absolute value of \j/(z) remains greater than a positive number M in this neighborhood. Denoting by r the absolute value of 2 a, we have |/()| >l//r m , and therefore |/(s)| becomes infinite when r approaches zero. Since the function \fr(z) is regular for z == a, there exists a circle C with the center a in the interior of which i/r() is analytic. The quotient \j/(z)/(z a) m is an analytic function for all the points of this circle except for the point a itself. In the neigh- borhood of a pole a, the function /() has therefore no other singulai point than the pole itself; in other words, poles are isolated singular points* 90 THE GENERAL CAUCHY THEORY [II, 41 41 Functions analytic except for poles Every function which is analytic at all the points of a legion A, except only for singular points that are poles, is said to be analytic except for poles in that region* A function analytic in the whole plane except for poles may have an infinite number of poles, but it can have only a finite number in any finite region of the plane The proof depends on a geneial theorem, which we must now recall If in a finite region A of the plane there exist an infinite number of points possessing a particular propei ty> there exists at least one limit point in the region A or on its boundary (We mean by limit point a point in every neighborhood of which there exist an infinite numbei of points possessing the given propeity) This proposition is pioved by the process of successive subdivisions that we have employed so often For bievity, let us indicate by (E) the assemblage of points con- sidered, and let us suppose that the region A is divided into squares, or portions of squares, by parallels to the axes Ox, Oy There will be at least one region A 1 containing an infinite number of points of the assemblage (E) By subdividing the region A l in the same way, and by continuing this process indefinitely, we can form an infinite sequence of regions J 1? A 2 , - , A n , that become smaller and smaller, each of which is contained in the preceding and contains an infinite number of the points of the assemblage. All the points of A n approach a limit point Z lying in the interior of or on the bound- ary of A. The point Z is necessarily a limit point of (#), since there are always an infinite number of points of (.E) in the interior of a circle having Z for center, however small the radius of that circle may be. Let us now suppose that the function f(z) is analytic except for poles in the mteiior of a finite legion A and also on the boundary r of that region. If it has an infinite number of poles in the region, it will have, by the preceding theorem, at least one point Z situated in A or on P, in every neighborhood of which it will have an infinite number of poles. Hence the point Z can be neither a pole nor an ordinary pomt. It is seen in the same way that the function f(z) can have only a finite number of zeros in the same region. It follows that we can. state the following theorem : Every function analytic except for poles in a finite region A and on its boundary has in that region only a finite number of zeros and only a finite number of poles. * Such fanctaoas are said by some writers to be meromorphic. TRANS. H, 42] SINGULAR POINTS 91 In the neighborhood of any point a, a function f(z) analytic except for poles can be put in the form (32) /(*) = (*-*)**, where <f> (z) is a regular function not zero f or z = a The exponent fji is called the order of f(z) at the point a The order is zero if the point a is neither a pole nor a zero for f(z) , it is equal to m if the point a is a zero of oider m for f(z), and to n if a is a pole of ordei n for /(). 42. Essentially singular points. Every singular point of a single- valued analytic function, which is not a pole, is called an essen tially singular point An essentially singular point a is isolated if it is possible to describe about a as a center a circle C in the interior of which the function f(z) has no other singular point than the point a itself; we shall limit ourselves for the moment to such points Laurent's theorem furnishes at once a development of the func- tion/^) that holds in the neighboihood of an essentially singular point Let C be a circle, with the center a, in the interior of which the function /(#) has no other singular point than a , also let c be a cucle concentric with and ulterior to C. In the circular ring included between the two circles C and c the function f(z) is analytic and is therefore equal to the sum of a series of positive and negative powers of z a, (33) /()= 4.(-a)-. m= eo This development holds true for all the points interior to the circle C except the point a, for we can always take the radius of the circle c less than \ a\ for any point z whatever that is different from a and lies in C f . Moreover, the coefficients A m do not depend on this radius ( 37) The development (33) contains first a part regular at the point a, say P(z a), formed by the terms with positive exponents, and then a series of terms in powers of l/(s a), This is the principal part of /(#) in the neighborhood of the singular point This principal part does not reduce to a polynomial in (& _ #)-i ; for the point z = a would then be a pole, contrary to the 92 THE GENERAL CAUCHY THEORY [II, 42 hypothesis * It is an integral transcendental function of l/(z a) In fact, let r be any positive number less than the radius of the circle C; the coefficient A_ m of the series (34) is given by the expression (37) the integral being taken along the circle C 1 with the center a and the radius r We have, then, (35) \A_ m \ where M(r) denotes the maximum of the absolute value of /() along the circle C' The series is then conveigent, provided that | & a \ is greater than r, and since r is a number which we may suppose as small as we wish, the series (34) is conveigent for every value of & different from a, and we can write where P(& a) is a regular function at the point a, and #[!/( a)] an integral transcendental functiont of l/(s a) When the absolute value of * & approaches zero, the value of /() does not approach any definite limit. More precisely, i/ a circle C is described with the point a as a center and with an arbitrary radius p, there always exists in the interior of this circle points zfor which /() differs as little as we please from any number given in advance (WBIBBSTBASS) Let us first prove that, given any two positive numbers p and M, there exist values of z for which both the inequalities, | # a \ < p, [/(#)! > M, hold. For, if the absolute value of /(#) weie at most equal to M when we have \z a\ < /o, 3fC(r) would be less than or equal to M for r < p, and, from the inequality (35), all the coeffi- cients A_ m would be zero, for the product c^fT(y)r m ^Mr would approach zero with r Let us consider now any value A whatever If the equation f(z) = A has roots within the circle C, however small the radius p * To avoid overlooking any hypothesis, it would he necessary to examine also the case m which the development of /(z) in the interior of O contains only positive powers of 2- a, the value /(a) of the function at the point a heing different from the term independent of z a in the series The point z- a would he a point of dzscorir tinuity for/(2) We shall disregard this kind of singularity, which is of an entirely artificial character (see helow, Chapter IV). f We shall frequently denote an integral function of a; by G(x) II, 42] SINGULAR POINTS 93 may be, the theorem is proved If the equation /(?) = A does not have an infinite number of roots in the neighboihood of the point a, we can take the radius p so small that in the interior of the circle C with the radius p and the center a this equation does not have any roots. The function < (2) = l/[/(s) A~\ is then analytic foi every point & within C except for the point a ; this point a cannot be any- thing but an essentially singular point for <(s), for otherwise the point would be either a pole or an ordinary point for/(s). There- foie, from what we have just proved, there exist values of z in the interior of the circle C for which we have |4>(*)|>7 or |/(*) however small the positive number may be This property sharply distinguishes poles from essentially singu- lar points. While the absolute value of the function /(#) becomes infinite in the neighborhood of a pole, the value of /() is completely indeterminate for an essentially singular point Picard * has demonstrated a more precise proposition by showing that every equation f(z) = A has an infinite number of roots in the neighboihood of an essentially singular point, theie being no excep- tion except for, at most, one particular value of A. Example The point 2 = is an essentially singular point for the function It is easy to prove that the equation e i/z = A has an infinite number of roots with absolute values less than />, however small p may be, provided that A is not zero Setting A = r (cos B + i sin 0), we deuve from the preceding equation z We shall have \z\ < p, provided that . There are evidently an infinite number of values of the integer k which satisfy this condition In this example there is one exceptional value of A, that is, A 0. But it may also happen that there are no exceptional values , such is the case, for example, for the function sm (1/e), near 2 = *JLnndle$ de rgcote Normale suptrieure, 1880, 94 THE GENERAL CAUCHY THEORY [II, 43 43. Residues. Let a be a pole or an isolated essentially singular point of a f unction /(s) Let us consider the question of evaluating the integral ff(z) d along the circle C drawn in the neighboihood of the point a with the center a The regular part P(& a) gives zero in the integration As for the principal part [l/(s a)], we can integrate it term by teim, for, even though the point a is an essentially singulai point, this series is uniformly convergent The integral of the general term is zero if the exponent m is greater than unity, for the primitive function A_ m /[(m l)(s a)" 1 "" 1 ] takes on again its original value after the variable has described a closed path If, on the con- trary, w = l, the definite integral A^fdz/fa a) has the value 2 TrzJLi, as was shown by the previous evaluation made in 34 We have then the result ~ 2iriX_ l = I /(*)<&, t/(O which is essentially only a particular case of the formula (23) for the coefficients of the Laurent development. The coefficient JLi is called the residue of the function f(z) with respect to the singular point a Let us consider now a function /(#) continuous on a closed boundary curve r and having m the interior of that curve T only a finite number of singular points a, b, c } - , L Let A,B,C, , L be the corresponding residues ; if we surround each of these singular points with a circle of very small radius, the integral //(#)<#, taken along r in the positive sense, is equal to the sum of the integrals taken along the small curves in the same sense, and we have the very important formula (36) C f()<fc=*27ri(A+B+C+ Jcr> which says that the integral ff(z)dz, taken along T in the positive sense, is equal to the product of 2m and the sum of the residues with respect to the singular points off(&) within the curve P It is clear that the theorem is also applicable to boundaries r com- posed of several distinct closed curves. The importance of residues is now evident, and it is useful to know how to calculate them rapidly. If a point a is a pole of order m for f(z), the product (z a) m f(z) is regular at the point <z, and the residue of f(z) is evidently the II, 44] APPLICATIONS OF THE GENERAL THEOREMS 95 coefficient of (2 a) m ~ l in the development of that pioduct. The rule "becomes simple in the case of a simple pole ; the residue is then equal to the limit of the product (z a)f(z) for * = a. Quite fre- quently the f unction f(z) appears under the form where the functions P(z) and Q(z) are regular for z = a, and P(a) is different from zero, while a is a simple zero for Q(z) Let Q(s) = (2 a)R(z), then the residue is equal to the quotient P(a)/R (a), or again, as it is easy to show, to P(a)/Q'(a). Ill APPLICATIONS OF THE GENERAL THEOREMS The applications of the last theorem are innumerable. We shall now give some of them which are related particularly to the evalua- tion of definite integrals and to the theory of equations. 44. Introductory remarks. Let f(z) be a function such that the product (& a)f(z) approaches zero with \z a\ The integral of this function along a circle y, with the center a and the radius p, approaches zero with the radius of that circle. Indeed, we can write / /Cy) If vj is the maximum of the absolute value of (z a)f(z) along the circle y, the absolute value of the integral is less than 2 Try, and con- sequently approaches zero, since t\ itself is infinitesimal with p. We could show in the same way that, when the product (& <j)f(%) approaches zero as the absolute value of z a becomes infinite, the integral j[ C) /()^j taken along a circle C with the center a, ap- proaches zero as the radius of the circle becomes infinite. These statements are still true if, instead of integrating along the entire circumference, we integrate along only a part of it, provided that the product (v a)/(s) approaches zero along that part. Frequently we have to find an upper bound for the absolute value of a definite integral of the form f a b f(x) dx, taken along the axis of reals. Let us suppose for definiteness a < b. We have seen above ( 25) that the absolute value of that integral is at most equal to the integral / a 6 |/(a) | dx, and, consequently, is less than M(b a) HM is an upper bound of the absolute value of 96 THE GENERAL CAUCHY THEORY [II, 45 45. Evaluation of elementary definite integrals The definite inte- gral J**F(x)fa, taken along the real axis, where F(x) is a rational function, has a sense, provided that the denominate! does not vanish for any real value of x and that the degree of the numeiator is less than the degree of the denominator by at least two units. With the origin as center let us describe a cncle C with a radius R large enough to include all the roots of the denominator of J?(), and let us considei a path of integration formed by the diameter A, traced along the real axis, and the semicncumference C', lying above the real axis. The only singular points of F(z) lying in the interior of this path are poles, which come from the roots of the denominator of F(%) for which the coefficient of i is positive Indicating by 3R L the sum of the residues relative to these poles, we can then write C F(*)d*+ f F JR J(&) As the radius R becomes infinite the integral along C" approaches zero, since the product *F(*) is zero for * infinite ; and, taking the limit, we obtain /*+ J-co We easily reduce to the preceding case the definite integrals ,*** I F(smx,cosx)dx, Jo where F is a rational function of sin x and cos x that does not become infinite for any real value of x } and where the integral is to be taken along the axis of reals Let us first notice that we do not change the value of this integral by taking for the limits x and & + 2 7T 3 where X Q is any real number whatever It follows that we can take for the limits TT and + TT, for example Now the classic change of variable tan (ar/2) = t reduces the given integral to the integral of a rational function of t taken between the limits oo and + oo, for tan (x/2) increases from QO to + oo when x increases from - TT to + TT We can also proceed in another way. By putting e** = & we have dx = d/i& 9 and Euler's formulae give 0080? = ;:; > Sin X = -^-; 2s 2 is II, 46] APPLICATIONS OF THE GENERAL THEOREMS 97 so that the given integral takes the form / J 2z As for the new path of integration, when x increases from to 2 TT the vanable # describes in the positive sense the ciicle of unit radius about the origin as center It will suffice, then, to calculate the resi- dues of the new rational function of z with respect to the poles whose absolute values are less than unity Let us take for example the integral / 27r ctn [(# a bi)/2]dx, which has a finite value if b is not zero. We have , fx a bi ctn /a a fo\ e^-fe- ( 1* H^^F or ctn ' ~ ,-. - b + c Hence the change of variable e = # leads to the integral f *-I-I-T' u/(O The function to be integrated has two simple poles and the corresponding residues are 1 and +2. If b is positive, the two poles are in the interior of the path of integration, and the integral is equal to 2 iri\ if b is negative, the pole & = is the only one within the path, and the integral is equal to 2 m The pro- posed integral is therefore equal to 2 iri, according as b is posi- tive or negative. We shall now give some examples which are less elementary. 46. Various definite integrals. Example 1 The function & mss /(l + z*) has the two poles + 1 and z, with the residues e~ m /2 % and e/^ % Let us suppose for defimteness that m is positive, and let us consider the boundary formed by a large semicircle of radius R about the origin as center and above the real axis, and by the diameter which falls along the axis of reals In the interior of this boundary the function e""*/^ + 2 2 ) has the single pole z = $, and the integral taken along the total boundary is equal to irer m . Now the integral along the semicircle approaches zero as the radius E becomes infinite, for the absolute value of the product ze 1 */^ + z*) along that curve approaches zero. Indeed, if we replace z by JR (cos 8 + % sin 0), we have 98 THE GENERAL CAUCHY THEORY [n, 46 and the absolute value e- 3 ** 1 * 6 remains less than unity when vanes from to TT As for the absolute value of the factor z/(l + 2 2 ), it approaches zero as z becomes infinite We have, then, m the limit 'dx = x 2 If we replace e mix by cos mx + * sm mx, the coefficient of % on the left-hand side is evidently zeio, for the elements of the integral cancel out in pairs Since we have also cos ( mx) = cos mx, we can write the preceding formula in J& the form (37) * cos mx , TT dx = - 1 1 + x 2 2 PIG 17 with the radii R and.r, and the straight lines AB, B*A' We have, then, the relation Example 2 The function e z /z is analytic in the interior of the bound- ary ABMB'A'NA (Fig 17) formed by the two semicircles BMB', A'NA, described about the origin as center -R which we can write also in the form /Zffix^e-tx r 0* r &z X I Z I Z ~~ c/(jBJfJB f ) J(A f NA) When r approaches zero, the last integial approaches in , we have, in fact, &z i where P(z) is a regular function at the origin, so that r *. ./(*,' * The integral of the regular part P (z) becomes infinitesimal with the length of the path of integration , as for the last integral, it is equal to the variation of Log (z) along A'NA, that is, to m The integral along BMB' approaches zero as R becomes infinite Tor if we put z = R (cos B + % sin 0), we find JL OZ = , } 2 and the absolute value of this integral is less than r e -.R8iii0tf0 = 2 C Jo Jo II, 46] APPLICATIONS OF THE GENERAL THEOREMS 99 When increases from to w/2, the quotient sm 9/0 decreases from 1 to 2/ir, and we have hence which establishes the proposition stated above Passing to the limit, we have, then (see I, 100, 2d ed.), e tar_ e -t or r JQ / -f- 00 i sma; _ _ v Jo ~ ~*' Example 3 The integral of the integral transcendental function e- 2 along the boundary OABO formed by the two radii OA and OJ5, making an angle of 45, and by the arc of a circle AB (Fig 18), is equal to zero, and this fact can be expressed as follows C e-^dx + C Jo J( = C J(O When the radius E of the circle to which the arc AB belongs becomes infinite, the in- tegial along the arc AB approaches zero In fact, if we put z = E [cos (0/2) + i sm (0/2)], that integral becomes FIG. 18 and its absolute value is less than the integral i/. 1 --"-** As in the previous example, we have 7? / ^ f 2 Jo B (-. 2 Jo The last integral has the value and approaches zero -when. K becomes infinite. 100 THE GENERAL CAUCHY THEORY [II, 46 Along the radius OB we can put z = p [cos(n-/4) + i sm(ir/4)], which gives e-* 2 = e-*P 2 , and as E becomes infinite we have at the limit (see I, 135, 2d ed , 134, 1st ed ) r +to */ T ir\, r + ,, VTT / e-*P 2 (cos- + &sm-)(fy> = \ e-^dx^-, Jo \ 4 4//o 2 or, again, / + 2J Vir/ TT TT\ I e-*P z dp = - ( cos- - z sin-) Jo 2 \ 4 4/ Equating the real parts and the coefficients of *, we obtain the values of 3?resnel's integrals, (38) 47, Evaluation of T (p) 17(1 ^). The definite integral Jo "*"*/> 1 where the variable x and the exponent p are real, has a finite value, provided that p is positive and less than one , it is equal to the product r (p) T (1 p) * In order to evaluate this integral, let us consider the function z* ~ l /(l + z), which has a pole at the point z = 1 and a branch point at the point z = Let us consider the boundary abmb'a'na (Fig 19) formed by the two circles. C and C', described about the origin with the radii r and p re- spectively, and the two straight lines ab and a'6', lying as near each other as we please above and below a cut along the axis Ox The function 2p-i/(l + z) is single-valued within this boundary, which contains only 19 one singular point, the pole z = 1 In order to calculate the value of the integral along this path, we shall agree to take for the angle of z that one which lies between and 2ir If R denotes the residue with respect to the pole z = 1, we have then The integrals along the circles and C' approach zero as r becomes infinite and as p approaches zero respectively, for the product z*/(l + z) approaches zero m either case, since <p < 1. * Replace t by 1/(1 + a) in the last formula of 135, Vol I, 2d ed , 134, 1st ed The formula (39), derived by supposing p to be real, is correct, provided the real part ofp lies between and 1 II, 48] APPLICATIONS OF THE GENERAL THEOREMS 101 Along ab, 2 is leal For simplicity let us replace 3 by # Since the angle of z is zeio along a&, ZP~ I is equal to the numeiical \alue of XP~ I Along afb f also z is leal, but since its angle is STT, we have The sum of the two integrals along ab and along If of therefore has for its limit -de. n g2wi(p i)"j i ^ Jo 1 + x The residue .K is equal to (- l)p-i, that is, to ec^-i)"*, if TT is taken as the angle of 1 We have, then, 1 + a or, finally, (39) /*+ ^-1 ^ v Jo 1 + x " ^ smjpir 48. Application to functions analytic except for poles. Given two functions, f(z) and < (^), let us suppose that one of them, /(), is analytic except for poles in the interior of a closed curve C, that the other, <f> (z), is everywhere analytic within the same curve, and that the three functions f(z), f r (z), <f> (%) aie continuous on the curve C , and let us try to find the smgulai points of the function < (%)/'(%)//(*) within C A point a which is neither a pole nor a zero for /(#) is evidently an ordinary point for the function f l (&)/f(z) and conse- quently for the function $ (#)/ f ()//(#) If a point a is a pole or a zeio of(z), we shall have, in the neighborhood of that point, where p, denotes a positive or negative integer equal to the order of the function at that point ( 41), and where $(%) is a regular func- tion which is not zero for & = a Taking the logarithmic derivatives on both sides, we find Since, on the other hand, we have, in the neighborhood of the point a, it follows that the point a is a pole of the first order for the product ^ (*)/'(*)//(*)> and its residue is equal to f*4(a)> tliat 1S > to w*(a), if the point a is a zero of order m for /(*), and to n<f> (a) if the point a is a pole of order n for/(s). Hence, by the general theorem 102 THE GENERAL CAUCHY THEORY [II, 46 of residues, provided there are no roots of f(z) on the curve O, we have 2^1 f c c where a is any one of the zeros of f(&) inside the boundary C, I any one of the poles of f(z) within C, and where each of the poles and zeros are counted a number of times equal to its degiee of multi- plicity The formula (40) furnishes an infinite iiumbei of relations, since we may take for < (z) any analytic function Let us take in particular <j> (z) = 1 , then the preceding formula becomes (41) *-p where N and P denote respectively the number of zeros and the number of poles of f(z) within the boundary C This formula leads to an important theorem In fact, f'(is)/f(&) is the denvative of Log [/(#)] ? to calculate the definite integial on the right-hand side of the formula (41) it is therefore sufficient to know the variation of log |/(*)| +' angle [/()] when the variable # describes the boundary C 111 the positive sense But |/() | returns to its initial value, while the angle of /(#) increases by 2 -ffTT, K being a positive or negative integer We have, therefore, (42) N - P that is, the difference N P is equal to the quotient obtained "by the division of the variation of the angle off(z) by 2 TT when the variable % describes the boundary C in the positive sense Let us separate the real part and the coefficient of i *&/(&) When the point z = x + yi describes the curve C in the positive- sense, the point whose coordinates are X, Y, with respect to a system of rectangular axes with the same orientation as the first system, describes also a closed curve C f 1 , and we need only draw the curve C l approximately in order to deduce from it by simple inspection the integer K In fact, it is only necessary to count the number of revolutions which the radius vector joining the origin of coordinates to the point (X, Y) has turned through in one sense or the other. II, 49] APPLICATIONS OF THE GENERAL THEOREMS 103 We can also write the formula (42) in the form /A*\ Ar D (43) N-P = Since the function F/Z takes on the same value after z has described the closed curve C, the definite integral / A<?3 - YdX is equal to irI(Y/X\ where the symbol I (Y/X) means the index of the quotient Y/X along the boundary C, that is, the excess of the number of times that that quotient becomes infinite by passing from + 00 to oo over the number of times that it becomes infinite by passing from oo to + oo (I, 79, 154, 2d ed , 77, 154, 1st ed.). We can write the formula (43), then, in the equivalent form (44) tf-P-| 49 Application to the theory of equations. When the function f(z) is itself analytic within the curve C, and has neither poles nor zeros on the curve, the preceding formulae contain only the roots of the equation /(#) = which lie within the region bounded by C. The formulae (42), (43), and (44) show the number N of these roots by means of the variation of the angle of /(#) along the curve or by means of the index of Y/X. If the function f(z) is a polynomial in 3, with any coefficients whatever, and when the boundary C is composed of a finite number of segments of umcursal curves, this index can be calculated by ele- mentary operations, that is, by multiplications and divisions of polynomials. In fact, let AB be an arc of the boundary which can be represented by the expressions where <j>(t) and \jf(t) are rational functions of a parameter t which varies from a to j8 as the point (#, y) describes the arc AB m the positive sense. Eeplacing 2 by (<)+ ty(9 m tlie polynomial /(*), we have -/ x where R (t) and R l (tf) are rational functions of t with real coefficients. Hence the index of Y/X along the arc AB is equal to the index of the rational function RJR as t vanes from a to ft which we already 104 THE GENERAL CAUGHT THEORY [n, 49 know how to calculate (I, 79, 2d ed , 77, 1st ed ) If the bound- ary C is composed of segments of unicursal curves, we need only ' calculate the index for each of these segments and take half of their sum, m oider to have the numbei of roots of the equation /() = within the boundaiy C Note D'Alembeit's theorem is easily deduced from the pieceding results Let us piove first a lemma which we shall have occasion to use seveial tunes Let F(%) 9 <(#) be two functions analytic in the interior of the closed cuive C, continuous on the curve itself, and such that along the entire curve C we have |$(s) | <\F(si) \ , under these conditions the two equations JF(*)=0, F()+*()=0 have the same number of roots, in the interior of C. For we have As the point z describes the boundary C, the point Z = 1 + $ (z)/F(z) describes a, closed curve lying entirely within the circle of unit radius about the point Z = 1 as center, since | Z 1 1 < 1 along the entire curve C Hence the angle of that factor returns to its initial value after the variable has described the boundary (7, and the variation of the angle of F(z) + <(s) is equal to the variation of the angle of F(z) Consequently the two equations have the same number of roots in the intenor of C 'Now let /() be a polynomial of degree m with any coefficients whatever, and let us set let us choose a positive number R so large that we have 4. *o *" Then along the entire circle (7, described about the origin as center with a radius greater than R, it is clear that |$/F| < 1 Hence the equation f(z) == has the same number of roots in the interior of the circle C as the equation F(z) = 0, that is, m. 50. Jensen's formula Let/(z) be an analytic function except for poles m the interior of the circle C with the radius r about the ongin as center, and ana- lytic and without zeros on C Let a x , a 2 , - , a* be the zeros, and & 1? & 2 , , b m the poles, of f(z) in the interior of this circle, each being counted according to its degree of multiplicity We shall suppose, moreover, that the origin is neither II, 50] APPLICATIONS OF THE GENERAL THEOREMS 105 a pole noi a zero for/(a) Let us evaluate the definite integral (45) I taken along C in the positive sense, supposing that the variable z starts, foi example, from the point z = r on the real axis, and that a definite determina- tion of the angle of f(z) has heen selected in advance Integrating- by parts, we have (46) I = {Log (a) Log [/()] } m - J^Log (*) |j cZz, where the first part of the right-hand side denotes the increment of the product Log (z) Log [f(z)1 when the variable z describes the circle C. If we take zero for the initial value of the angle of z, that increment is equal to (log r + 2 m) {Log [/(r)] + 2 TTI (n - m)} - log r Log [/(r)] = 27Tt Log [/(r)] + 2Tri(n m)logr 4(n- T^Tr 3 In order to evaluate the new definite integral, let us consider the closed curve T, formed by the circumference (7, by the circumference c described about the origin with the infinitesimal radius p, and by the two borders a&, #?>' of a cut made along the real axis from the point z = p to the point z = r (Fig 19) We shall suppose for definiteness that f(z) has neither poles nor zeros on that portion of the axis of reals If it has, we need only make a cut making an infinitesimal angle with the axis of reals The function Logs is analytic in the interior of F, and according to the general formula (40) we have the relation The integral along the circle c approaches zero with /o ? for the product z Log z is infinitesimal with p On the other hand, if the angle of z Is zero along ct6, it is equal to 2ir along <&'&', and the sum of the two corresponding integrals has for limit o f(z) The remaining portion is and the formula (46) becomes (n- m)logr + 2 TTI Log [/(O)] -2mLogs 4(n, - In order to integrate along the circle (7, we can put s = r&4 and let <0 vary from to 2 ir It follows that dz/z = id<p Let/() = JRe*#, where B and * are 106 THE GENERAL CAUCHY THEORY [II, 50 continuous functions of 4> along C Equating the coefficients of i in the preced- ing relation, we obtain Jensen's formula * (47) _ r log Bfy = log |/(0) | + log x ' in "which there appear only ordinary Napierian logarithms. When the function /(z) is analytic in the interior of 0, it is clear that the product ^ - & should be replaced by unity, and the formula becomes (48) f "log JB d$ = log |/(0) | + log 27T JO This relation is interesting m that it contains only the absolute values of the roots of f(z) within the circle C, and the absolute value of f(z) along that circle and for the center of the same circle 51 . Lagrange's formula. Lagrange's formula, which we have already established by Laplace's method (I, 195, 2d ed , 189, 1st ed ), can be demonstrated also very easily by means of the general theorems of Cauchy. The process which we shall use is due to Hermite Let /(#) be an analytic function in a certain region D containing the point a. The equation ^ ^ (49) jr(*)=*-^ + /(*)=0, where a is a variable parameter, has the root % a, f or a = 1 Let us suppose that a ^ 0, and let C be a circle with the center a and the radius r lying entirely in the region D and such that we have along the entire circumference o/(s)| < |at a|. By the lemma proved in 49 the equation F(z) has the same number of loots within the curve C as the equation a = 0, that is, a single loot Let f denote that root, and let II (&) be an analytic function in the circle C. The function U(z)/F() has a single pole m the interior of C, at the point 2 = f , and the corresponding residue is n()/F'(). From the general theorem we have, then, = JL C n(*)<fo = i 2J<(7) **() 3 *ri In order to develop the integral on the right in powers of a, we shall proceed exactly as we did to derive the Taylor development, * Acta rnatihematica, Vol XXII t It is assumed that/(a) is not zero, for otherwise F(z) would vanish when 2 a for any value of a and the following developments would not yield any results of interest TRANS. II, 31] APPLICATIONS OF THE GENERAL THEOREMS 107 and we shall write , . X* - a)-* 1 * - a - /() * - *J Substituting this value in the irrtegial, we find. where r _ ~" -*-- /() - a Let m be the maximum value of the absolute value of of (si) along the circumfeience of the circle C, then, by hypothesis, m is less than r If M is the maximum value of the absolute value of II (*) along C, we have which shows that JK n ^ approaches zero when n increases indefinitely. Moreover, we have, by the definition of the coefficients J" , /^ - - -, / n , . . and the formula (14), whence we obtain the following development in series : We can write this expression in a somewhat different form If we take n ()=/f r () [1 V a/()], "where *(*) is an analytic function in the same region, the left-hand side of the equation (50) will no longer contain a and will reduce to *() As for the right-hand side, we observe that it contains two terms of degree n in a, whose sum is {.we/cm- 108 THE GENERAL CAUCIIY THEORY [II, 51 \ and we find again Lagrange's formula in its usual form (see I, formula (52), 195, 2d ed ; 189, 1st ed ) O>n 3n 1 (51) **() + f '()/()+ +^^3lW)[/( a )] w }+ ' - We have supposed that we have \af(z)\<r along the cncle C, which is true if \a is small enough. In order to find the maximum value of | a |, let us limit ourselves to the case where /(*) is a poly- nomial or an integral function Let M(r) be the maximum value of \f() \ along the cncle C described about the point a as center with the radius r The proof will apply to this circle, provided \a\M(r) <r We are thus led to seek the maximum value of the quotient r/M(r), as r varies from to + oo This quotient is zero for r = 0, for if &C(r) weie to approach zero with r, the point z = a would be a zero for /(), and F(z) would vanish foi = a. The same quotient is also zeio for r = oo, for otherwise /() would be a polynomial of the first degree ( 36) Aside from these trivial cases, it follows that r/$C(r) passes through a maximum value p for a value r x of r The reasoning shows that the equation (49) has one and only one root such that | a\<r l9 provided |a|</&. Hence the developments (50) and (51) are applicable so long as \a\ does not exceed /*, pro- vided the functions H() and &(&) are themselves analytic in the circle C t of radius r r Example. Let/(z) = (z 2 l)/2 , the equation (49) has the root 1 Vl 2aa+ a* f = - a - ' which approaches a when a approaches zero Let us put II (z) = 1. Then the formula (50) takes the form where X n is the nth Legendre's polynomial (see I, 90, 189, 2d ed , 88, 184, 1st ed ) In order to find out between what limits the formula is valid, let us suppose that a is real and greater than unity On the circle of radius r we have evidently #T(r) = [(a + r) 2 1]/2, and we are led to seek the maximum value of 2r/[(a + r) 2 1] as r increases from to + > This maximum is found for r = Va 2 1, and it is equal to a Va 2 1 If, however, a lies between 1 and + 1, we find by a quite elementary calculation that The maximum of 2rVl cP/(i* + 1 a 2 ) occurs when r = Vl a 2 , and it is equal to unity. II, 52] APPLICATIONS OF THE GENERAL THEOREMS 109 It is easy to verify these results In fact, the ladical Vl-- 2aa+ a: 2 , con- sidered as a function of or, has the two critical points a Va 2 1 If a > 1, the critical point nearest the origin is a Va 2 1 "When a lies between 1 and + 1, the absolute value of each of the two critical points a i Vl a 2 is unity In the fourth lithographed edition of Hermite's lectures will be found (p 185) a very complete discussion of Kepler's equation z a = smz by this method His piocess leads to the calculation of the root of the transcendental equation e r (r 1) = er r (r + 1) which lies between 1 and 2 Stieltjes has obtained the values fj. = 1 ,199678640257734, p = 6627434193492 52 Study of functions for infinite values of the variable. In order to study a function /() for values of the variable for which the absolute value becomes infinite, we can put & = 1/z* and study the function /(!/') in the neighborhood of the origin But it is easy to avoid this auxiliary transformation We shall suppose first that we can find a positive number R such that every finite lalue ofz whose absolute value is greater than R is an ordinary point for/(#) If we descube a circle C about the origin as center with a radius R, the function /(#) will be regular at every point z at a finite distance lying outside of C. We shall call the region of the plane exterior to C a neighborhood of the point at infinity. Let us consider, together with the circle C, a concentiic circle C' with a radius R' > R. The function f(&), being analytic in the circular ring bounded by C and C f , is equal, by Laurent's theorem, to the sum of a series arranged according to integral positive and negative powers of 2, (53) /(*)= A_ m *; 7ft= 00 the coefficients A_ m of this series are independent of the radius R 1 , and, since this radius can be taken as large as we wish, it follows that the formula (53) is valid for the entire neighborhood of the point at infinity, that is, for the whole region exterior to C We shall now distinguish several cases : 1) When the development of f(z) contains only negative powers of , (54) f^)=^ + A^+A^ 3 +... + A m ^ + ..., the f unction /(#) approaches A^ when \z\ becomes infinite, and we say that the function f(z) is regular at the point at infinity) or, again, that the point at infinity is an ordinary point for f(z). If the 110 THE GENERAL CAUCHY THEORY [II, 52 coefficients A Q , A I} , A m _ 1 are zero, but A m is not zeio, the point at infinity is a zero of the rath order for f(z). 2) When the development of f(z) contains a finite number of positive powers of z, (55) /(*)=*^ + <B-i*"- 1 + where the first coefficient B m is not zero, we shall say that the point at infinity is a pole of the wth order for /(*?), and the polynomial B m z m + - + B^z is the principal part relative to that pole. When \z\ becomes infinite, the same thing is tiue of |/(s)|, whatever may be the manner in which z moves 3) Finally, when the development of f(z) contains an infinite number of positive powers of #, the point at infinity is an essentially singular point for f(&) The series formed by the positive powers of z represents an integral function G(z), which is the principal part in the neighborhood of the point at infinity. We see in particular that an integral transcendental function has the point at infinity as an essentially singular point The preceding definitions were in a way necessitated by those which have already been adopted for a point at a finite distance Indeed, if we put z = /z' } the function f(z) changes to a function of z', <t> (V)==/(l/*Oj an< ^ ^ 1S seen a ^ once ^ na/ k we k ave on ty carried over to the point at infinity the terms adopted for the point z' == with respect to the function <f> (#'). Eeasoning by analogy, we might be tempted to call the coefficient A_ l of z, in the development (53), the residue, but this would be unfortunate In order to preserve the characteristic property, we shall say that the residue with respect to the point at infinity is the coefficient of l/# with its sign changed, that is, A r This number is equal to where the integral is taken in the positive sense along the boundary of the neighborhood of the point at infinity. But here, the neighbor- hood of the point at infinity being the part of the plane exterior to C, the corresponding positive sense is that opposite to the usual sense. Indeed, this integral reduces to II, 32] APPLICATIONS OF THE GENERAL THEOREMS 111 and, when 2 describes the circle C in the desired sense, the angle of diminishes by 2 TT, which gives A as the value of the integral It is essential to observe that it is entirely possible for a function to be regular at the point at infinity without its residue being zero , for example, the function 1 + 1/s has this property. If the point at infinity is a pole or a zero for /(), we can write, in the neighborhood of that point, where ft is a positive or negative integer equal to the order of the function with its sign changed, and where < () is a function which is regulai at the point at infinity and which is not zero for = oo. Fiom the preceding equation we deduce where the function <p'(z)/<l>(z) is regular at the point at infinity but has a development commencing with a term of the second or a higher degiee in 1/s. The residue of f(x)/f(z) is then equal to /*, that is, to the older of the f unction /(s) at the point at infinity. The state- ment is the same as for a pole or a zero at a finite distance. Let/(#) be a single-valued analytic function having only a finite number of singular points. The convention which has just been made for the point at infinity enables us to state in a very simple form the following general theorem The sum of the residues of the function f(z) in the entire plane, the point at infinity included, is &ero. The demonstration is immediate Describe with the origin as center a circle C containing all the singular points of /(*) (except the point at infinity) The integral //() dz, taken along this circle in the oidinary sense, is equal to the product of 2 iri and the sum of the residues with respect to all the singular points of f(z) at a finite distance On the other hand, the same integral, taken along the same circle in the opposite sense, is equal to the product of 2 iri and the residue relative to the point at infinity. The sum of the two integrals being zero, the same is true of the sum of the residues. Cauchy applied the term total residue (residu integral) of a func- tion /(#) to the sum of the residues of that function for all the singular points at a finite distance. When there are only a finite number of singular points, we see that the total residue is equal to the residue relative to the point at infmity with its sign changed. 112 THE GENERAL CAUCHY THEORY [II, 52 Example. Let * where P( and Q() are two polynomials, the first of degree p, the second of even degree 2q. If 22 is a real number greater than the absolute value of any root of Q(, the function is single-valued out- side of a circle C of radius R, and we can wxite where <(>) is a function which is regular at infinity, and which is not zero for * = oo. The point at infinity is a pole for/() iS.p>q 9 and an ordinary point ifp&q. The residue will certainly be zero if p is less than q 1. IV. PERIODS OF DEFINITE INTEGRALS 53* Polar periods. The study of line integrals revealed to us that such integrals possess periods under certain circumstances Since every integral of a function /(*) of a complex variable * is a sum of line mtegials, it is clear that these integrals also may have certain periods. Let us consider first an analytic function /(*) that has only a finite number of isolated singular points, poles, or essentially singular points, within a closed curve C. This case is absolutely analogous to the one which we studied for line integrals (I, 153), and the reasoning applies here without modification Any path that can be drawn within the boundary C between the two points & , Z of that region, and not passing through any of the singular points of /(*), is equivalent to one fixed path joining these two points, preceded by a succession of loops starting from z^ and surrounding one or more of the singular points a v a^ * , a n of /() Let A v A^ , A+ be the corresponding residues of /(*) ; the integral //(*)<&, taken along the loop surrounding the point a l9 is equal to 2 mA l9 and similarly for the others. The different values of the integral fff(*)d& are therefore included in the expression r* (56) I /(*) d = F(Z) + 2 Tri (m l A l + m 2 A 2 + + m n A n ), J** where F(Z) is one of the values of that integral corresponding to the determined path, and m t , m 2 , - are arbitrary positive or nega- tive integers , the periods are n, 53] PERIODS OF DEFINITE INTEGRALS 113 In most cases the points a v a^ , a n are poles, and the periods result from infinitely small ciicuits described about these poles , whence the teim polar periods, which is oidmarily used to distin- guish them from peiiods of another kind mentioned later. Instead of a region of the plane interior to a closed curve, we may consider a portion of the plane extending to infinity , the function f(z) can then have an infinite number of poles, and the integral an infinite number of peiiods. If the residue with respect to a singu- lar pomt a of f(z) is zero, the corresponding period is zero and the point a is also a pole or an essentially singular point for the integral. But if the residue is not zero, the point a is a logarithmic critical point for the integral. If, for example, the point a is a pole of the mth order for/(s), we have in the neighborhood of that point and therefore R . +3 1 Log( a) where C is a constant that depends on the lower limit of integration * and on the path followed by the variable in integration When we apply these general considerations to rational functions, many well-known results are at once apparent. Thus, in order that the integral of a rational function may be itself a rational function, it is necessary that that integral shall not have any periods ; that is, all its residues must be zero. That condition is, moreover, sufficient The definite integral C z d has a single critical point z = a, and the corresponding period is 2 Tri , it is, then, in the integral calculus that the true origin of the multiple values of Log(s a) is to be found, as we have already pointed out in detail in the case of ffdz/z ( 31). Let us take, in the same way, the definite integral dz r C Jo it has the two logarithmic critical points + i and i, but it has only the single period TT. If we limit ourselves to real values of the 114 THE GENERAL CAUCHY THEORY [II, 53 variable, the diffeient determinations of arc tana; appear as so many distinct functions of the vanable x We see, on the contrary, how Cauehy's woik leads us to regard them as so many distinct branches of the same analytic function Note "When there aie more than three periods, the value of the definite integral at any point z may be entirely indeterminate Let us recall first the following result, taken from the theory of continued fractions* Given a real irrational numbei or, we can always find two integers p and <?, positive or nega- tive, such that we have \p + qa\ < e, where e is an arbitianly preassigned positive number The numbers p and q having been selected in this way, let us suppose that the sequence of multiples of p + qa is formed Any real number A is equal to one of these multiples, or lies between two consecutive multiples We can therefore find two integers m and n such that | m + no: A\ shall be less than e, With this in mind, let us now consider the function 2 m \2 a zb z c z d)' where a, &, c, d are four distinct poles and or, /3 are leal irrational numbers The integral f z *f(z)dz has the four periods 1, <z, i, ifi. Let I(z) be the value of the integral taken along a particular path from z to z, and let M + Ni denote any complex number whatevei We can always find f oui integers m, n, m', n' such that the absolute value of the diffeience na + %(mf + n'p) will be less than any preassigned positive number e We need only choose these integers so that , 2 where M + Ni I(z) A + Bi Hence we can make the variable describe a path joining the two points given in advance, 2 , 2, so that the value of the inte- gral ff(z) dz taken along this path differs as little as we wish from any pre- assigned number Thus we see again the decisive influence of the path followed by the variable on the final value of an analytic function 54. A study of the integral f*dz/-Vl z 2 The integral calculus explains the multiple values of the function arc sin & in the simplest manner by the preceding method They arise from the different determinations of the definite integral according to the path followed by the variable. For defimteness we shall suppose that we start from the origin with the initial value + 1 * A little farther on a direct proof will be f ound ( 66) II, 54] PERIODS OF DEFINITE INTEGRALS 115 for the radical, and we shall indicate by I the value of the integral taken along a determined path (01 direct path) Foi example, the path shall be along a straight line if the point # is not situated on the real axis 01 if it lies upon the real axis within the segment from 1 to + 1 , but when z is real and \\ > 1, we shall take for the duect path a path lying above the real axis Now, the points =+l,s= 1 being the only critical points of Vl 3 , every path leading fiom the origin to the point can be xeplaced by a succession of loops described about the two critical points 4- 1 and 1, followed by the direct path. We are then led to study the value of the mtegial along a loop. Let us considei, for example, the loop OamaO, described about the point & = + 1 ; this loop is composed of the segment Oa passing from the origin to the point 1 c, of the cucle ama of radius c described about # = 1 as center, and of the segment aO Hence the integral along the loop is equal to the sum of the integials dx The integral along the small circle approaches zero with c, for the product (z 1) f(z) approaches zero. On the other hand, when z has described this small circle, the radical has changed sign and in the integral along the segment aO the negative value should be taken for Vl x 2 The integial along the loop is therefore equal to the limit of 2f Q l ~ e dx/^/l a? as e approaches zero, that is, to TT It should be observed that the value of this integral does not depend on the sense in which the loop is described, but we return to the origin with the value 1 for the radical If we were to describe the same loop around the point *? = + 1 with 1 as the initial value of the radical, the value of the integral along the loop would be equal to TT, and we should return to the origin with + 1 as the value of the radical. In the same way it is seen that a loop described around the critical point # = 1 gives TT or + TT f or the integral, according as the initial value + 1 or 1 is taken for the radical on starting from the origin. If we let the variable describe two loops in succession, we return to the origin with + 1 for the final value of the radical, and the value of the integral taken along these two loops will be + 2 TT, 0, or 116 THE GENERAL CAUCHY THEORY [n, 54 2 TT, according to the oider in which these two loops are descubed An even nuinbei of loops will give, then, 2 mir foi the value of the integral, and will bung back the ladical to its initial value +1 An odd number of loops will give, on the contiary, the value (2 m -f 1) TT to the integial, and the final value of the ladical at the origin will be 1 It follows from this that the value of the integial F(z) will be one of the two forms according as the path described by the variable can be replaced by the dnect path preceded by an even number or by an odd number of loops 55. Periods of hyperelliptic integrals. We can study, in a similar manner, the different values of the definite integral (58) where P (&) and R (#) are two polynomials, of which the second, R (& of degree n, vanishes for n distinct values of z We shall suppose that the point # is distinct from the points e v e 2 , , e n ; then the equation w 2 = R (Z Q ) has two distinct roots + and U Q We shall select w for the initial value of the radical R () If we let the variable & descube a path of any form whatever not pass- ing through any of the critical points e^ e 2 , , e n , the value of the radical V.R() at each point of the path will be determined by con- tinuity Let us suppose that from each of the points e l9 e 2 , , e n we make an infinite cnt in the plane in such a way that these cuts do not cross each other The integral, taken from up to any point z along a path that does not cross any of these cuts (which we shall call a direct path), has a completely determined value I(z) for each point of the plane. We have now to study the influence of a loop, described from around any one of the critical points e t , on the value of the integral Let 2 E % be the value of the integral taken along a closed curve that starts from and incloses the single criti- cal point e v the initial value of the ladical being w . The value of this integral does not depend on the sense in which the curve is described, but only on the initial value of the radical at the point . In fact, let us call 2 E{ the value of the integral taken along the same II, 35] PERIODS OF DEFINITE INTEGRALS 117 curve in the opposite sense, with the same initial value II Q of the radical If we let the variable # describe the curve twice in succes- sion and in the opposite senses, it is clear that the sum of the inte- grals obtained is zero ; but the value of the integial for the first turn is 2 E l} and we return to the point 2 with the value U Q for the radi- cal The integral along the curve described in the opposite sense is then equal to 2 E^ 9 and consequently E( = E % . The closed curve considered may be reduced to a loop formed by the straight line #, the circle c % of infinitesimal radius about e l} and the straight line az Q (Fig. 21) , the integral along c t is infinitesimal, since the pioduct (# e t ) P (z)/-*jR(z) approaches zero with the absolute value of n 0,. If we add together the integrals along z Q a and along az Q , we find 21 where the integral is taken along the straight line and the initial value of the radical is This being the case, the inte- gral taken along a path which reduces to a succession of two loops described about the points e a , e$ is equal to 2 J2* 2 Ep 9 for we return after the first loop to the point # with the value U Q for the radical, and the integral along the second loop is equal to 2Ep After having described this new loop we return to the point # with the original initial value U Q If the path described by the variable # can be reduced to an even number of loops described about the points e a , e& e y , e S) - -, e K , e K successively, followed by the direct path from # to #, where the indices a, ft, , K, \ are taken from among the numbers 1, 2, , n, the value of the integral along the path is, by what precedes, +2(E K - EJ. JF(*) = 1+ 2(E a - J If, on the contrary, the path followed by the variable can be reduced to an odd number of loops described successively around the critical points e a , e^ , e K , e^, e^ the value of the integral is 118 THE GENERAL CAUCHY THEOKY [II, 55 Hence the integral under consideiation has as periods all the expres- sions 2(E, - E h ), but all these periods reduce to (n 1) of them for it is clear that we can mite 2(E t - E h }=2(E t - E Since, on the othei hand, 2 E^ = v + 2 E n , we see that all the values of the definite integral F(z) at the point are given by the two expressions where m i; w 2 , , w n _! are arbitiary integers This lesult gives use to a certain number of impoitant observa- tions. It is almost self-evident that the penods must be independent of the point 3 chosen for the starting point, and it is easy to verify this Considei, for example, the penod 2E % ~-~2 E h ; this period is equal to the value of the integral taken along a closed curve r pass- ing through the point Z Q and containing only the two cutical points e l} e h . If, for defimteness, we suppose that there are no other critical points in the interior of the triangle whose vertices are # , e %9 e h , this closed curve can be replaced by the boundary bb'nc'emb (Fig 21) ; whence, making the radii of the two small circles approach zero, we see that the period is equal to twice the integral taken along the straight line joining the two critical points t , e h . It may happen that the (n 1) periods t^, w 2 , - , (o n-1 are not independent This occurs whenever the polynomial R (2) is of even degree, provided that the degree of P(z) is less than n/2 1. With the point * as center let us draw a circle C with a radius so large that the circle contains all the critical points , and for simplicity let us suppose that the critical points have been numbered from 1 to n in the order in which they are encountered by a radius vector as it turns about in the positive sense The integral taken along the closed boundary ^AMA^ formed by the radius # Q A, by the circle (7, and by the radius Az Q described in the negative sense, IU55] PERIODS OF DEFINITE INTEGRALS 119 is zero The integrals along %^A and along Az Q cancel, for the circle C contains an even number of critical points, and after having described this circle we return to the point A with the same value of the radical On the other hand, the integral along C approaches zero as the radius becomes infinite, since the product zP(z)/^R(z) approaches zero by the hypothesis made on the degree of the poly- nomial P(z) Since the value of this integral does not depend on the radius of C, it follows that that value must be zero. Now the boundaiy z Q AMAz Q considered above can be replaced by a succession of loops described around the critical points e l9 e^ * - - , e n in the order of these indices Hence we have the relation which can be written in the form ! <* 2 + <* <4+ + <-! ==0; and we see that the n 1 periods of the integral reduce to n 2 peuods oij, 2 , , o) n _ 2 Consider now the more general form of integral = Jz n where P, Q, E are three polynomials of which the last, E(z), has only simple roots. Among the roots of Q (z) there may be some that belong to E (z) , let or 1T o: 2 , , <x g be the roots of Q (z) which do not cause E (z) to vanish. The integral F(z) has, as above, the periods 2(J t -Z^), where 2 E t denotes always the inte- gral taken along a closed curve starting from z and inclosing none of the roots of either of the polynomials Q(z) and E(z) except e, But F(z) has also a cer- tain number of polar periods arising from the loops described about the poles <ar p (3r a , -, a t The total number of these periods is again diminished by unity if E (z) is of even degree n, and if where p and q are the degrees of the polynomials P and Q respectively Example Let E (z) be a polynomial of the fourth degree having a multiple root Let us find the number of periods of the integral If E (z) has a double root e l and two simple roots e 2 , %, the integral dz 120 THE GENERAL CAUCHY THEORY [il, 55 has the period 2 E z 2J 3 , and also a polar penod arising fiom a loop aiound the pole e l By the remark made ]ust above, these two periods are equal If E (z) has two double loots, it is seen immediately that the integral has a single polar period If E (z) has a tuple root, the integral dz has the period 2 jE^ 2 J 2 , but, by the general remaik made above, that period is zeio The same thing is tiue if E(z) has a quadiuple loot In resume' we have If E (z) has one or two double roots, the integi al has a pet lod , if E (z) has a tnple or quadruple root, the integral does not have periods All these results are easily venfied by direct integration 56. Periods of elliptic integrals of the first kind. The elliptic integral of the first kind, where E (z) is a polynomial of the third or the fourth degree, prime to its derivative, has two periods by the preceding general theory We shall now show that the ratio of these two periods ^s not real We can suppose without loss of generality that R() is of the third degree Indeed, if R^z) is a polynomial of the fourth degree, and if a is a root of this polynomial, we may write (I, 105, note, 2ded ; 110, 1st ed) where & = a + 1/y and where R (y) is a polynomial of the third degree It is evident that the two integrals have the same periods If E(z) is of the third degree, we may suppose that it has the loots and 1, for we need only make a linear substitution = a + py to reduce any other case to this one Hence the proof reduces to showing that the integral (59) F(z) ' W , V*(l -*)(*-*) where a is different from zero and from unity, has two periods whose ratio is not real If a is real, the property is evident Thus, if a is greater than unity, for example, the integral has the two periods c i ** 2 Jo Vs(l-3)(a-3)' J II, 56] PERIODS OF DEFINITE INTEGRALS 121 of which the first is real, while the second is a pure imaginary. Moreover, none of these periods can be zero. Suppose now that a is complex, and, for example, that the coeffi- cient of i in a is positive. We can again take for one of the penods dz We shall apply Weierstrass's formula ( 27) to this integral. When * varies from to 1, the factor l/Vg(l z] remains positive, and the point representing I/ V# z describes a curve L whose general nature is easily determined. Let A be the point representing a ; when z varies from to 1, the point a & descubes the segment AB parallel to Ox and of unit length (Fig 22). Let Op and Oq be the bisectors of the angles which the straight lines OA and OB make with Ox, and let Op 1 and Oq 1 be straight lines sym- metrical to them with respect to Ox. If we select that determination of Va z whose angle lies between and 7T/2, the point V& z de- FIG 22 scribes an are aft from a point a on Op to a point /3 on Oq , hence the point I/ V# 3 describes an arc #'' from a point #' on Op 1 to a point ft' of 0#'. It follows that Weierstrass's formula gives where ^ is the complex number corresponding to a point situated in the interior of every convex closed curve containing the aic a 1 ft 1 . It is clear that this point Z l is situated in the angle p'Og', and that it cannot be the origin , hence the angle of Z^ lies between 7r/2 and 0. We can take for the second period d* o C = 9, I V*(l-*)(a-*) Jo or, setting z = atf, 2 122 THE GENERAL CAUCHY THEORY [II, 56 In order to apply Weierstrass's formula to this integral, let us notice that as t increases from to 1 the point at describes the segment OA and the point 1 at describes the equal and parallel segment from s = 1 to the point C Choosing suitably the value of the radical, we see, as before, that we may write /*(! - where Z^ is a complex number different from zero wJwse angle lies between and -rr/2 The latio of the two periods ^/fy or 2jZ 1 is therefore not a real number. EXERCISES 1, Develop the function in powers of , m being any number Emd the radius of the circle of convergence 2 Find the different developments of the function l/[(z 3 + 1) (z 2)] in posi- tive or negative powers of 2, according to the position of the point z in the plane 3 Calculate the definite integral /2 2 Log[(z + l)/(z - l)]dz taken along a circle of radius 2 about the origin as center, the initial value of the logarithm at the point 2 = 2 being taken as real Calculate the definite integral dz taken over the same boundary 4 Let f(z) be an analytic function in the interior of a closed curve C con- taining the origin Calculate the definite integral j^f^LogzcZs, taken along the curve C, starting with an initial value z 6 Derive the relation f +j dt __ 135* (2n-l) i~ 246 -2w and deduce from it the definite integrals r y oo 6 . Calculate the following definite integrals by means of the theory of residues . -^ ~-i m and a being real, c($ 2 -h a 2 ) 2 r-j Jo x r / 00 cos OKB , , . -daj, a being real, n,Exs] EXERCISES 123 + * L r , a and p being real, cosxdx "JT J + ao jclogg(%c r Jo (l + <") 8 Jo 3 cos ax cos & dte, a and 6 being real and positive (To evaluate the last integral, integrate the function (eP lz Pw)/s 2 along the boundary indicated by Fig 17 ) 7 The definite integral f Q v d<p/[A + C ( J. C) cos 0] is equal, when it has any finite value, to eir/^/AC, where e is equal to 1 and is chosen in such a way that the coeflicient of t in ei VAC /A is positive 8 Let 2^(2) and (z) be two analytic functions, and 2 = a a double root of G(z) = that is not a root of F(z) Show that the corresponding residue of F(z)/G(z) is equal to 6 ff'(q) G"(a) - 2 F(a) G'"(a) In a similar manner show that the residue of F(z)/[G(z)f for a simple root a of G (z) = is equal to F'(a)G'(a)-F(a)G"(a) [G'(a)]* 9 Derive the formula the integral being taken along the real axis with the positive value of the radical, and a being a complex number or a real number whose absolute value is greater than unity. Determine the value that should be taken for Vl a 2 10 Consider the integrals J^cfe/Vl + 2 8 , J^dz/Vl + z*, where S and S^ denote two boundaries formed as follows The boundary S is composed of a straight-line segment OA on Ox (\vhich is made to expand indefinitely), of the circle of radius OA about as center, and finally of the straight line AO. The boundary S l is the succession of three loops which inclose the points a, 6, c which represent the roots of the equation z 5 + 1 = 0. Establish the relation that exists between the two integrals /1 + x* which arise in the course of the preceding consideration 11 By integrating the function e-* 8 along the boundary of the rectangle formed by the straight lines y = 0, y = 6, x = + E, x = 12, and then making R become infinite, establish the relation er *? cos 2 bx dx = 124 THE GENERAL CAUCHY THEORY [II,Exs 12. Integrate the function tr z &~\ where n is real and positive, along a boundary formed by a ladius OA placed along Ox, by an aic of a circle AB of radius OA about as center, and by a radius SO such that the angle a. = AOB lies between and w/2. Making OA become infinite, deduce from the preced- ing the values of the definite integrals C w w n-i e -att COS & u d f ^-^e-^sin&ttdw, Jo Jo where a and b are real and positive The results obtained are valid for a = ?r/2, provided that we have n < 1 13 Let m, m', 71 be positive integers (m < w, ra' < n) Establish the formula - T /2m +1 \ . /2m" + 1 VI t = ctnl TT ctn( ir] 2nL \ 2n / \ 2n ] 14 Deduce from the preceding result Euler's formula /' Jo 15. If the real part of a is positive and less than unity, we have f */ 1 + & sin air (This can be deduced from the formula (39) ( 47) or by integrating the function e as /(l + c a ) along the boundary of the rectangle formed by the straight lines ?/ = 0, 2/ = 2?r, = +E, x= JB, and then making E become infinite ) 16 Derive in the same way the relation -dec = TT (ctn aw ctn &?r), where the real parts of a and 6 are positive and less than unity (Take for the path of integration the rectangle formed by the stiaight lines y = 0, y = TT, x = E, x = 12, and make use of the preceding exercise ) 17 Erom the formula C J((7 where n and & are positive integers, and G is a circle having the origin as center, deduce the relations ( 2 co SW )+icos( TO - *)<! = ,( + !)( + ) ( + *) / + 1 x* n dx 1 3 6 (2n-l) ,_ x /1-JZJ 2 246 2n (Put 2 = e 2w , then cos w = $, and replace n by n + fc, and A; by n.) 18*. The definite integral II, Exs ] EXERCISES 125 when it has a finite value, is equal to TT/ V 1 2 <xx -f a: 2 , where the sign depends upon the relative positions of the two points a and x Deduce from this the expression, due to Jacobi, for the nth Legendre's polynomial, = - C If /0 19. Study in the same way the definite mtegial f Jo J x a + \ x 2 1 cos ^ and deduce from the result Laplace's formula o where e = 1, according as the real part of x is positive or negative 20* Establish the last result by integrating the function 1 along a circle about the origin as center, -whose radius is made to become infinite. 21* Gauss's sums. Let T s e**/*, where n and s are integers , and let S n denote the sum T Q + T : + + T n _i Deuve the formula 2 (Apply the theorem on residues to the function &(z) = e 2msZ/n /(e' 2 ' tnz 1), talang for the boundary of integration the sides of the rectangle formed by the straight lines x = 0, x = n, y = + JS, y = E, and inserting two semicircumf erences of radius e about the points x = 0, x = n as centers, in order to avoid the poles 2 = and z = n of the function <j> (z) , then let It become infinite ) 22 Let/(z) be an analytic function in the interior of a closed curve T con- taming the points a, 6, c, - , I If ar, j5, - , X are positive integers, show that the sum of the residues of the function fey with respect to the poles a, 5, c, , I is a polynomial F(x) of degree *v i * a + /3+... + X-l satisfying the relations J"(a) =/(a), . \ *C- (a) =/<- =/(6) f , ^ (Make use of the relation ^(x) = /(x) + [/^ <j> (z) dz]/2 m ) 23* Let/(2) be an analytic function m the interior of a circle C with center a On the other hand, let a x , o^, , c^, - - be an infinite sequence of points within the circle C, the point a n having the center a for limit as n becomes in- finite. Eor every point z within C there exists a development of the form 126 THE GENERAL CAUCHY THEORY [II, Exs where F n (z) = (z~a 1 )(z-a 2 ) (*-o) [LAURENT, Journal de mathematiques, 5th series, Vol VIII, p 325 ] (Make use of the following formula, which is easily verified, z x z - ! (z ajte- a 2 ) (s-aO (a-fln-i) } . 1 (s-fli) -(s-g*^ - - ) (z a,,) ' and follow the method used m establishing Taylor's foimula ) 24 Let z = a + bi be a root of the equation f(z) = X + Ti = of multi- plicity w, where the function f(z) is analytic m its neighborhood The point x=a, y = b is a multiple point of order n for each of the two cuives JC = 0, F = The tangents at this point to each of these cuives form a set of lines equally inclined to each other, and each ray of the one bisects the angle between the two adjacent rays of the othei 25 Let /(a) = X + Yi = A Q z + A^- 1 + + A m be a polynomial of the mth degree whose coefficients aie numbers of any kind All the asymptotes of the two curves X = 0, Y pass through the point -4 1 /mJL and are arranged like the tangents in the preceding exercise 26* Burman's series. Given two functions /(#), F(x) of a variable x, Burman's formula gives the development of one of them in powers of the other To make the problem more definite, let us take a simple root a of the equation F(x) = 0, and let us suppose that the two functions /(x) and F(x) are analytic in the neighborhood of the point a In this neighborhood we have '"-'is- the" function 4>(x) being regular for JG = a if a is a simple root of F(x) = Representing F(x) by y, the preceding relation is equivalent to x a ?/0 (x) = 0, and we are led to develop /(x) in powers of y (Lagrange's formula) 27*. Kepler's equation The equation z a e sin z 0, where a and e are two positive numbers, a < ir, e < 1, has one leal loot lying between and -rr, and two roots whose leal parts he between mir and (m + !)TT, wheie m is> any positive even mtegei or any negative odd mtegoi If m is positive and odd, or negative and even, there aie no roots whose leal parts he between mir and [BRIOT ET BOUQUET, TMone desfonctions ellyptiques, 2d ed , p 199 ] (Study the curve descubed by the point u = z a esinz when the van- able z describes the four sides of the rectangle formed by the stiaight lines x = mTT, x =s (m + 1) TT, y = + #, y = JK, where E is very large ) 28* Foi very large values of m the two roots of the preceding exercise whose real parts lie between 2 WTT and (2 m + 1) TT are approximately equal to 7T/2 * [log (2/e) + log (2 mir + Tr/2)] [COURIER, Annafa de VJScole Normale, 2d series, Vol VII, p 73.] CHAPTER III SINGLE-VALUED ANALYTIC FUNCTIONS The first part of this chapter is devoted to the demonstration of the general theoiems of Weierstiass* and of Mittag-Leffler on inte- gral functions and on single-valued analytic functions with an infinite number of singular points TTe shall then make an applica- tion of them to elliptic functions Since it seemed impossible to develop the theory of elliptic func- tions with any degree of completeness in a small number of pages, the treatment is limited to a general discussion of the fundamental principles, so as to give the reader some idea of the importance of these functions. For those who wish to make a thorough study of elliptic functions and their applications a simple course in Mathe- matical Analysis would never suffice ; they will always be compelled to turn to special treatises. I. WEIERSTRASS'S PRIMARY FUNCTIONS. MITTAG-LFFLER*S THEOREM 57. Expression of an integral function as a product of primary functions. Every polynomial of the rath degree is equal to the prod- uct of a constant and m equal or unequal factors of the form x a, and this decomposition displays the roots of the polynomial. Euler was the first to obtain for sin z an analogous development in an infinite product, but the factois of that product, as we shall see fai- ther on, are of the second degree in a. Cauchy had noticed that we are led in certain cases to ad-join a suitable exponential factor to each of the binomial factors such as x a. But Weierstrass was the first to treat the question with complete generality by showing that every integral function having an infinite number of roots can be expressed as the product of an infinite number of factors, each of which vanishes for only a single value of the variable. * The theorems of Weierstrass which are to be presented here were first published in a paper entitled Zur Theone, der eindeutzgen analyttschen Functionen (Berl Abhandlungen, 1876, p 11 = Werke, Vol II, p 77). Picard gave a translation of this paper in the AnnaUs de PEcote Normale superwure (1879) The collected researches of Mittag-Leffler are to be found in a memoir m the Acta mathematics Vol II 127 128 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 57 We alieady know one integral function which does not vanish for any value of , that is, e?. The same thing is true of e ff(z > 9 where g(z) is a polynomial or an integral transcendental function Conversely, every integral function which does not vanish for any value of 2 is expressible in that foim In fact, if the integral function G(z) does not vanish for any value of 2, every point & = a is an ordinary point for G'(z)/G(z) } which is therefore another integral function g^z) : ) Integrating both sides between the limits # , #, we find where g(z) is a new integral function of 2, and we have G(z) = G(zJtf<*- ff W = 0*C*>-^+ L si: <?(*<,>]. The right-hand side is precisely in the desired form. If an integral function G (&) has only n roots a v 2 , , a n , distinct or not, the function G (2) is evidently of the form Let us consider now the case where the equation G(z)=Q has an infinite number of roots. Since there can be only a finite number of roots whose absolute values are less than or equal to any given num- ber R ( 41), if we arrange these roots in such a way that their absolute values never diminish as we proceed, each of these roots appears in a definite position in the sequence where \a n \ ^ |a m+1 |, and where \a n \ becomes infinite with the index n. We shall suppose that each root appears in this series as often as is required by its degree of multiplicity, and that the root & = is omitted from it if <?(0)=: We shall first show how to construct an integral function 6^(2) that has as its roots the numbers in the sequence (1) and no others. The product (1 z/a n }e Q v&, where Q, v (z) denotes a polynomial, is an integral function Vhich does not vanish except for & = a n We shall take for Q v (z) a polynomial of degree v determined in the fol- lowing manner : write the preceding product in the form in, 57] PRIMARY FUNCTIONS 129 and replace Log (1 /a w ) by its expansion in a power series , then the development of the exponent will commence with a term of degree v + 1, piovided we take The integer v is still undetermined We shall show that this number v can be chosen as a function of n in such a way that the infinite product will be absolutely and uniformly convergent in every circle C of radius R about the origin as center, however large R may be The radius R having been chosen, let a be a positive number less than unity. Let us consider separately, m the product (2), those factors corresponding to the roots a n whose absolute values do not exceed R/a If there are % roots satisfying this condition, the product of these factors evidently represents an integral function of Consider now the product of the factors beginning with the (# + l)th : If % remains in the interior of the circle with the radius R, we have [#|^ JR, and since we have \a n \>R/a when n>q, it follows that we also have \\ <tf[a w |. A factor of this product can then be written, from the manner in which we have taken Q v (z), t g h_l. V < if we denote this factor by 1 + M M we have l/ay + l 1 /s\y-t-2 u n==e -T+i(%J M^W/ '1. Hence the proof reduces to showing that by a suitable choice of the number v the series whose general term is ?7 n = |^ w | is uniformly convergent in the circle of radius R (I, 176, 2d ed ). In general, if m is any real or complex number, we have 130 SINGLE-VALUED ANALYTIC FUNCTIONS [ui, 5? We liave then, a fortiori, 1 I Z |" + 1 /14.HI| 2 \4.?1\^\ 2 + A 7 n ^/+ilM \ l+ + al^r^-j-skl * ;_i ? or, noticing that || <a|a n |, when || is less than R, But if aj is a real positive number, eF 1 is less than cce*; hence we have l *~~ v-\~l a n 1 a """ v -J- 1 a tt 1 a In order that the series whose general term is U n shall be uni- formly convergent in the circle with the radius R, it is sufficient that the series whose general term is &/a n v+1 converge uniformly in the same circle If theie exists an integer p such that the series S|l/# n | y converges, we need only take v =p 1 If there exists no integer p that has this property,* it is sufficient to take v = n 1 For the series whose general term is z/a n \ n is uniformly convergent in the circle of radius R, since its terms are smaller than those of the senes 2,\R/a n \ n , and the nth root of the general term of this last series, or |-R/a n [, approaches zero as n increases mdefinitely.t Therefoie we can always choose the integer v so that the infinite product -F 2 (#) will be absolutely and uniformly convergent in the circle of radius R. Such a product can be replaced by the sum of a uniformly convergent series ( 176, 2d ed) whose terms are all analytic. Hence the pioduct F^(s) is itself an analytic function within this circle (39). Multiplying &%(&) by the product F a (), which contains only a finite number of analytic factors, we see that the infinite product is itself absolutely and uniformly convergent in the interior of the circle C with the radius JK, and represents an analytic function within this circle. Since the radius R can be chosen arbitrarily, and since * For example, let onlog n (w^2) The series whose general term is (log ri)-P w divergent, whatever may be the positive number p t for the sum of the first (n - 1) terms is greater than (n-l)/0ogn)p, an expression which becomes infinite with n f Borel has pointed out that it is sufficient to take for v a number such that v + 1 shall be greater than logra In fact, the senes S| J2/on|i<>& is convergent, for the general term can be written e^e* i<* I -R/i!s*w lo st /!. After a sufficiently large value of n, \a n \/K will be greater than &, and the general term less than 1/w 2 . Ill, 57] PRIMARY FUNCTION'S 131 v does not depend on J2, this product is an integral function 6^(2) which has as its roots precisely all the various numbers of the sequence (1) and no others. If the integral function G (z) has also the point z as a root of the pth order, the quotient is an analytic function which has neither poles nor zeros in the whole plane Hence this quotient is an integral function of the form e ff& y wheie g(z) is a polynomial or an integial tianscendental func- tion, and we have the following expression for the function G (2) (4) <?(*) = The integral function g(z) can in its tuin be replaced in an infinite variety of ways by the sum of a umfoiinly convergent series of polynomials and the preceding foiinula can be written again TT/ GW=*II{ -^ The factors of this product, each of which vanishes only for one value of 2, are called primary functions Since the product (4) is absolutely convergent, we can arrange the primary functions in an arbitrary order or group them together in any way that we please. In this product the polynomials Q v (z) depend only on the roots themselves when we have once made a choice of the law which determines the number v as a function of n. But the exponential factor e g< & cannot be determined if we know only the roots of the function G(z) Take, for example, the function sin TTZ, which has all the positive and negative integers for simple roots In this case the series S f |l/ n | 2 is convergent, hence we can take v = 1, and the function where the accent placed to the right of n means that we are not to give the value zero* to the index n, has the same roots as sin TT#. * When this exception is to be made in a formula, we shall call attention to it by placing an accent (0 after the symbol of the product or of the sum 1S2 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 57 We have then SHUT* = e? w G(), but the reasoning does not tell us anything about the factoi e 9(z) We shall show later that this factor reduces to the numbei IT 58. The class of an integral function. Given an infinite sequence a v a 2 , , tf n , *, where a n \ becomes infinite with n, we have just seen how to construct an infinite number of integial functions that have all the terms of that sequence for zeros and no others When there exists an integer^; such that the series Sl^l""* is convergent, we can take all the polynomials Q v (z) of degree p 1 Given an integial function of the form where P() is a polynomial of degree not higher than p 1, the number ^ 1 is said to be the class of that function Thus, the function is of class zero , the function (sin irz)/ i jr mentioned above is of class one. The study of the class of an integial function has given rise in recent years to a large number of investigations * 59. Single-valued analytic functions with a finite number of singular points. When a single-valued analytic function F(z) has only a finite number of singular points in the whole plane, these singular points are necessarily isolated; hence they are poles or isolated essentially singular points The point z = oo is itself an ordinary point or an isolated singular point ( 52) Conversely, if a single- valued analytic function has only isolated singular points in the entire plane (including the point at infinity), there can be only a finite number of them In fact, the point at infinity is an ordinary point for the function or an isolated singular point. In either case we can describe a circle C with a radius so large that the function will have no other singular point outside this circle than the point at infinity itself Within the circle the function can have only a finite number of singular points, for if it had an infinite number of them there would be at least one limit point ( 41), and this limit point would not be an isolated singular point Thus a single-valued analytic *See BOREI*, Lemons sur les fonctzons entieres (1900), and the recent work of BLTJMBNTHAL, Sur ksfonctions entieres de genre inflni (1910) Ill, 59] PRIMARY FUNCTIONS 133 function which has only jjoles has necessarily only a finite number of them, foi a pole is an isolated singulai point Every single-valued analytic function which is regular for every finite value of z, and for z = oo , is a constant In fact, if the func- tion were not a constant, since it is legular for every finite value of z, it would be a polynomial or an integral function, and the point at infinity would be a pole or an essentially singular point. 3STow let F(z) be a single-valued analytic function with n distinct singular points a v 2 , - ., a n in the finite portion of the plane, and let G t [/(z a,)] be the principal part of the development of F(z) in the neighborhood of the point a v , then G % is a polynomial or an integral transcendental function in l/(z a t ) In either case this principal part is regular for every value of z (including & = oo) except z == a l Similarly, let P(z) be the principal part of the devel- opment of F(z) in the neighborhood of the point at infinity. P(z) is zero if the point at infinity is an ordinary point for F(z). The difference is evidently regular for every value of z including z = GO ; it is there- fore a constant C, and we have the equality* (5) ^ ) = P(s)+t __ + c , which shows that the function F(&) is completely determined, except for an additive constant, when the principal part in the neighbor- hood of each of the singular points is known. These principal parts, as well as the singular points, may be assigned arbitrarily. When all the singular points are poles, the principal parts G l are polynomials; P(z) is also a polynomial, if it is not zero, and the right-hand side of (5) reduces to a rational fraction Since, on the other hand, a single-valued analytic function which has only poles for its singular points can have only a finite number of them, we conclude from this that a single-valued analytic function, all of whose singular points are poles, is a rational fraction. * We might obtain the same formula "by equating to zero the sum of the residues of the function where z and z are considered as constants and X as the variable (see 52). 134 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 60 60. Single-valued analytic functions with an infinite number of singu- lar points. If a single-valued analytic function has an infinite num- ber of singular points in a finite region, it must have at least one limit point within 01 on the boundaiy of the region For example, the function l/sm(l/s) has as poles all the roots of the equation sin (l/s)= 0, that is, all the points * = I/&TT, where k is any integer whatever The origin is a limit point of these poles Similarly, the function sin has for singular points all the roots of the equation sin (l/#) among which are all the points 2 = __, 2&'7r-j-aresin( J where k and k' are two arbitrary integers All the points l/(2 FTT) are limit points, for if, U remaining fixed, k increases indefinitely, the preceding expression has l/(2&'7r) for its limit It would be easy to construct more and more complicated examples of the same kind by increasing the number of sin symbols There also exist, as we shall see a little farther on, functions for which every point of a certain curve is a singular point. It may happen that a single-valued analytic function has only a finite number of singular points in every finite portion of the plane, although it has an infinite number of them in the entire plane. Then outside of any circle C, however great its radius may be, there are always an infinite number of singular points, and we shall say that the point at infinity is a limit point of these singular points. In the following paragraphs we shall examine single-valued analytic func- tions with an infinite number of isolated singular points which have the point at infinity as their only limit point. 61. Mittag-Leffler's theorem. If there are only a finite number of singular points in every finite portion of the plane, we can, as we have already noticed for the zeros of an integral function, arrange these singular points in a sequence (6) a l9 a 2 , - ., a n , in snch a way that we have \a n \ ^ \a n+l \ and that \a n \ becomes infinite with n* We may suppose also that all the terms of this sequence Ill, 61] PRIMARY FUNCTIONS 135 are different To each term a l of the sequence (6) let us assign a polynomial 01 an mtegial function in l/(s ,)> C?, [!/( a,)], taken in an entirely aibitraiy manner. Mittag-Lefflei's theorem may be stated thus There exists a single-valued analytic function which is regular for every finite value ofz that does not occur in the sequence (6), and for which the principal part in the neighborhood of the point z = a, is We shall prove this by showing that it is possible to assign to each function G t [l/( a t )] a polynomial P t () such that the series defines an analytic function that has these properties. If the point 2 = occurs in the sequence (6), we shall take the corresponding polynomial equal to zero. Let us assign a positive number e^ to each of the other points a % so that the series Sc, shall be convergent, and let us denote by a a positive nuinbei less than unity. Let C z be the circle about the origin as center passing through the point a lt and C[ the circle concentnc to the preceding with a radius equal to a\a t \. Since the function <7 t [l/( a 4 )] is analytic in the circle C l? we have for every point within C t The power series on the right is uniformly convergent in the circle C^ ? hence we can find an integer v so large that we have, in the interior of the circle C(, Having determined the number v in this manner, we shall take for P t (s) the polynomial <r l0 a*iZ a iv z v . How let C be a circle of radius R about the point z == as center. Let us consider separately the singular points a % in the sequence (6) whose absolute values do not exceed R/a. If there are q of them, we shall set 136 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 61 The remaining infinite series, is absolutely and uniformly convergent in the circle <7, since for every point in this circle \&\ < R < a|# t | if the index i is greater than # From the inequality (7), and from the manner in which we have taken the polynomials P % (z), the absolute value of the geneial term of the second series is less than ^ when # is within the circle C. Hence the function F^z) is an analytic function within this circle, and it is clear that if we add *\(z) to it, the sum (8) F(z) will have the same singular points in the circle C, with the same principal parts, as F^z). These singular points are precisely the terms of the sequence (6) whose absolute values are less than R, and the principal part in the neighborhood of the point a, is GJ[l/(z a t )]. Since the radius R may be of any magnitude, it follows that the function F(z) satisfies all the conditions of the theorem stated above. It is clear that if we add to F(v) a polynomial or any integral function whatever G(z), the sum F(z) -f G(z) will have the same singular points, with the same principal parts, as the function F(z) Conversely, we have thus the general expression for single-valued analytic functions having given singular points with corresponding given principal parts , for the difference of two such functions, being regular for every finite value of , is a polynomial or a transcendental integral function Since it is possible to represent the function G(z) in turn by the sum of a series of polynomials, the function F(z) + G (z) can itself be represented by the sum of a series of which each term is obtained by adding a statable polynomial to the principal part <?,[!/(* -a,)]- If all the principal parts G % are polynomials, the function is analytic except for poles in the whole finite region of the plane, and conversely. We see, then, that every function analytic except for poles can be represented by the sum of a series each of whose terms is a rational fraction which becomes infinite only for a single finite value of the variable. This representation is analogous to the decom- position of a rational fraction into simple elements Every function $(z) that is analytic except for poles can also be represented by the quotient of two integral functions. For suppose Ill, 62] PRIMARY FUNCTIONS 137 that the poles of $ (z) are the terms of the sequence (6), each being counted accoidmg to its degree of multiplicity Let G(z) be an mtegial function having these zeros, then the product (z) G(&) has no poles It is therefore an integral function 6^(2), and we have the equality 62. Certain special cases. The preceding demonstration of the general theorem does not always give the simplest method of con- structing a single-valued analytic function satisfying the desired conditions Suppose, foi example, it is required to construct a func- tion &(z) having as poles of the first order all the points of the sequence (6), each residue being equal to unity , we shall suppose that 2 = is not a pole. The principal part relative to the pole a, is l/(g a t ), and we can write _i ^!__ _^ + _l 2 "** v & If we take the proof reduces to determining the integer v as a function of the index i in such a way that the series -foo -. shall be absolutely and uniformly convergent in every circle de- scribed about the origin as center, neglecting a sufficient number of terms at the beginning For this it is sufficient that the series S(/a t ) v+1 be itself absolutely and uniformly convergent in the same region. If there exists a number^? such that the series S|l/a t |* is convergent, we need only take v=p 1. If there exists no such integer, we will take as above ( 57) v = i - 1, or v + 1 > log i. The number v having been thus chosen, the function (9) *(*) which is analytic except for poles, has all the points of the sequence (6) as poles of the first order with each residue equal to unity. 188 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 62 It is easy to deduce from this a new proof of Weierstrass's theorem on the decomposition of an integral function into primary functions In fact, we can integrate the series (9) term by term along any path whatever not passing through any of the poles , for if the path lies in a cucle C having its centei at the origin, the series (9) can be replaced by a series which is unifoimly convergent in this circle, together with the sum of a finite number of functions analytic except for poles This results from the demonstration of formula (9) If we integrate, taking the point z = Q for the lower limit, we find and consequently It is easy to verify the fact that the left-hand side of the equation (10) is an integral function of &. In the neighborhood of a value a of z that does not occur in the sequence (6) the integral f Q *& is analytic ; hence the function is also analytic and different from zero for z = a. In the neighbor- hood of the point ff t we have I Jo where the functions P and Q are analytic It is seen that this inte- gral function has the terms of the sequence (6) for its roots, and the formula (10) is identical with the formula (3) established above The same demonstration would apply also to integral functions hav- ing multiple roots If a % is a multiple root of order r, it would suffice to suppose that $(#) has the pole & = a % with a residue equal to r Let us try again to form a function analytic except for poles of the second order at all the points of the sequence (6), the princi- pal part in the neighborhood of the point a % being l/(z a^f. We shall suppose that z = is an ordinary point, and that the series I 8 is convergent, it is clear that the series S|l/<z t | 4 will also III, 63] PRIMARY FUNCTIONS 189 be convergent Limiting the development of !/(# a^ in powers of z to its first term, we can write and the series 2a l z-z^ A 2 ' satisfies all the conditions, provided it is uniformly convergent in every circle C descubed about the origin as center, neglecting a sufficient number of terms at the beginning ISTow if we take only those terms of the series coming from the poles a l for which we have \a l \> R/a, R being the radius of the circle C and a a positive num- ber less than unity, the absolute value of (1 z/a^)~ z will remain less than an upper bound, and the series whose general term is 2 z/a% 2 /a* is absolutely and uniformly convergent in the circle C, by the hypotheses made concerning the poles a z . 63. Cauchy's method. If F(z) is a function analytic except for poles, Mittag-Leffler's theorem enables us to form a series of rational terms whose sum F^z) has the same poles and the same principal parts as F(z) But it still remains to find the integral function which is equal to the difference F(z) F^z) Long before Weierstrass's work, Cauchy had deduced from the theory of residues a method by which a function analytic except for poles may, under very general condi- tions on the function, be decomposed into a sum of an infinite number of rational terms It is, moreover, easy to generalize his method Let F(z) be a function analytic except for poles and regular in the neighborhood of the origin ; and let C 19 C^ , C n? be an infinite succession of closed curves surrounding the point & = 0, not pass- ing through any of the poles, and such that, beginning with a value of n sufficiently large, the distance from the origin to any point what- ever of C n remains greater than any given nurnbei. It is clear that any pole whatever of F(z) will finally be interior to all the curves C n > c *+i> ' ' 9 provided the index n is taken large enough. The definite integral -J- where x is any point within C n different from the poles, is equal to F(x) increased by the sum of the residues of F(z)/(& x) with 140 SINGLE-VALUED ANALYTIC FUNCTIONS [ill, 63 respect to the different poles of F(z) within C n . Let a L be one of these poles. Then the conesponding principal pait G K \l/(z A )] is a rational function, and we have in the neighborhood of the point a k In the neighboihood of this point we can also write 1 = 1 = _ 1 _ Z-OL _ (z - atf Writing out the product we see that the residue of F(z)/( or) with respect to the pole a L is equal to __ ^1 ^-m-l ^m . We have, then, the relation where the symbol S indicates a summation extended to all the poles a k within the curve C n On the other hand, we can replace l/(s x) by aaid "write the preceding formula m the form ^ r o / 2 7T^ J ( c n (13) The integral is equal to F(0) increased by the sum of the residues of F(z)/z with respect to the poles of F(z) within C n . More generally, the definite integral 1 C 27r Vcc tt is equal to plus the sum of the residues of tr r F(&) with respect to the poles of F(z) within C n . If we represent by 4 r -*> the residue of z~ r F(z) m, es] PRIMARY FUNCTIONS 141 relative to the pole a L , -we can wiite the equation (13) m the form (14) *(<>) + ? _L. r m/* 2m J(c^^ x ^ In order to obtain an upper bound for the last term, let us -write it in the foim = ^i! C 2 *Ac n Let us suppose that along C n the absolute value of sr p F(z) remains less than M, and that the absolute value of z is greater than 8 Since the number n is to become infinite, we may suppose that we have already taken it so laige that 8 may be taken greater than |#|; hence along C n we shall have 1 ^ 1 If S n IB the length of the curve C n) we have then M* +1 We shall have proved that this term R n approaches zero as n becomes infinite if we can find a sequence of closed curves C v C z> > , C n> - and a positive integer^? satisfying the following conditions: 1) The absolute value of z~ p l r (ii) remains less than a fixed num- ber M along each of these curves. 2) The ratio S n /8 of the length of the curve C n to the minimum distance 8 of the origin to a point of C n remains less than an upper bound Z as n becomes infinite If these conditions are satisfied, \R n \ is less than a fixed number divided by a number 8 \x \ which becomes infinite with n. The term R n therefore approaches zero, and we have in the limit (15) Thus we have found a development of the function F(x) as a sum of an infinite series of rational terms The order in which they occur 142 SINGLE- VALUED ANALYTIC FUNCTIONS [III, 63 in the seiies is determined by tlie arrangement of the cuives CO- C in their sequence If the series obtained is abso- lutely 2 convergent, we can mite the teims in an arbitiary order Note. If the point = were a pole for -F(*0 with the pimeipal part <?(!/*), it would suffice to apply the preceding method to the function F(z) - G (I/*). 64. Expansion of ctn* and of sin* Let us apply this method to the function F() =ctn *!/*, which has only poles of the first oider at the points * = ITT, where k is any integer diif erent from zero, the residue at each pole being equal to unity We shall take for the curve C n a square, such as BCB'C', having the ougin for center and having sides of length 2nir + ir parallel to the axes , none of the poles are on this boundary, and the ratio of the length S n to the minimum distance S from the origin to a point of the boundary is constant and equal to 8. The squaie of the absolute value of ctn (x + yi) is equal to O nir On the sides BC and B'C 1 we have cos 2 a? = 1, and the absolute value is less than 1. On the sides BB' and CO 1 the square of this absolute value is less than FIG. 23 We must replace 2 y in this formula by (2 w + 1) TT, and the ex- pression thus obtained approaches unity when n becomes infinite. Since the absolute value of l/ along C n approaches zero when n becomes infinite, it follows that the absolute value of the function ctn 1/3 on the boundary C n remains less than a fixed number M, whatever n may be Hence we can apply to this function the for- mula (15), taking^ = 0. We have here ~ ,K T F(Q) = hm ^ / * x cos x sin x x since \ A ) = 0, / and 4 ? which represents the residue of (ctn 3 !/)/ kir, is eqiial to I/^TT. We have, then, (16) 1 n / -I -J > cta-ilimV'( V~ + r~ X noori w * 7r ;7r ' Ill, 64] PRIMARY FUNCTIONS 143 where the value k = is excluded from the summation The infinite series obtained by letting n become infinite is absolutely convergent, for the geneial teirn can be written in the form x LIT ATT A/TT (A-7T a;) and the absolute value of the factor a*/(l x/kir) remains less than a certain uppei bound, provided x is not a multiple of TT We have, then, precisely (IT) Integrating the two members of this relation along a path start- ing from the origin and not passing through any of the poles, we find from which we derive (18) am* The factor &&> is here equal to unity If in the series (17) we combine the two terms which come from opposite values of k, we obtain the formula Combining the two factors of the product (18) which correspond to opposite values of fc, we have the new formula* or, substituting irx for Note 1 The last f ormulsa show plainly the periodicity of sm x, which does not appear from the power series development. We see, in fact, that (sin wa;)/7r is the limit as n becomes infinite of the polynomial * This decomposition of sin x into an infinite product is due to Enler, who obtained at m an elementary manner (Introductto ^n Analy&in infimtorwn) SINGLE-VALUED ANALYTIC FUNCTIONS [III, 64 Replacing x by x + 1, this formula may be written in the form whence, letting n become infinite, we find sm (irx + TT) = sinine, or sin (2 + TT) = sins;, and therefore sin (z + STT) = sing Note 2. In this particular example it is easy to justify the necessity of associ- ating with each binomial factor of the form 1 or/a*, a suitable exponential factoi if we wish to obtain an absolutely convergent product For defimteness let us suppose x real and positive The series Zx/n being divergent, the product becomes infinite with m, while the product K) (-3 approaches zero as n becomes infinite (I, 177, 2d ed ) If we take m = n, the product P m Qm has (SHITTX)/*- for its limit , but if we make m and n become infinite independently of each other, the limit of this product is completely in- determinate This is easily verified by means of Weierstrass's primary functions, whatever may be the value of x Let us note first that the two infinite products are both absolutely convergent, and their product Fi(x)F 9 (x) is equal to (sixnnc)/7r. With these facts m mind, let us write the product P m Q n in the form When the two numbers m and n become infinite, the product of all the fac- tors on the right-hand side, omitting the last, has F T (x) F z (x) = (smirx)/ir for its limit. As for the last factor, we have seen that the expression has for its limit log , where w denotes the limit of the quotient m/n (I, 161) The product P m Q has, therefore, for its limit. Hence we see the manner m which that limit depends upon the law according to which the two numbers m and n become infinite. Note 3. We can make exactly analogous observations on the expansion of ctn x We shall show only how the periodicity of this function can be deduced from the series (17). Let us notice first of all that the series whose general term is Ill, 65] ELLIPTIC FUNCTIONS 145 where the index fc takes on all the integral values from oo to 4- *>, excepting k = 0, k = 1, is absolutely convergent , and its sum is 2/7r, as is &een on mat- ing k vary first f lorn 2 to + oo, then from 1 to oo "We can therefore write the development of ctnx in the form x - x J where the values fc = 0, k I are excluded from the summation This results from subtracting from each term of the series (17) the corresponding term of the convergent series formed by the preceding series togethei with the additional term 2/V Substituting x + IT f or sc, we find or, again, where fc 1 takes on all integral values except identical with etna. The light-hand side is II DOUBLY PERIODIC FUNCTIONS ELLIPTIC FUNCTIONS 65. Periodic functions. Expansion in senes. A single-valued analytic function /() is said to be periodic if there exists a real or complex number o> such that we have, whatever may be z,f(& + w)=j*(*)> this number o> is called a period. Let us mark in the plane the point representing <o, and let us lay off on the unlim- ited straight line pass- ing through the origin and the point o> a length _ equal to | | any number of times in both direc- tions. We obtain thus the points o>, 2 o>, 3 o>, - t w<o, - and the points CD, 2 o>, - , wo), .-. Through these different points and through the origin let us draw parallels to any direction differ- ent from Ov, the plane is thus divided into an infinite number of cross strips of equal breadth (Fig. 24) FIG 24 146 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 65 If through any point z we draw a parallel to the direction <9co, we shall obtain all the points of that straight line by allowing the leal parameter X in the expiession z 4- Xo> to vary from QQ to + o In particular, if the point z describes the fiist strip AA } BE^ the corre- sponding point z -f (o will describe the contiguous stup BB'CC 1 , the point 3 -f- 2 w will describe the thud strip, and so on in this manner. All the values of the f unction /(#) in the first strip will be duplicated at the corresponding points in each of the other strips Let LV and MM 1 be two unlimited straight lines parallel to the direction Oo Let us put u = e 2tirs ' u , and let us examine the region of the ^-plane described by the vanable u when the point z remains in the unlimited cross strip contained between the two parallels LL* and J/A/' If a -f /3ns a point of LL\ we shall obtain all the other points of that straight line by putting z = a + f$i + X<o and making X vary from oo to -f oo Thus, we have hence, as X varies from oo to -f co , u describes a circle C l having the origin for center Similaily, we see that as & describes the straight line J/fl/', u remains on a circle C 2 concentric with the first , as the point describes the unlimited strip contained between the two straight lines LV, MM*, the point u describes the ring-shaped region contained between the two circles C 19 C r But while to any value of # there corresponds only one value of u 9 to a value of u there correspond an infinite number of values of & which form an arithmetic progression, with the common difference o>, extending forever in both directions A periodic function /(), with the period CD, that is analytic in the infinite cross strip between the two straight lines LL 1 , MM*, is equal to a function <() of the new variable u which is analytic in the ring-shaped region between the two circles C^ and C 2 . For although to a value of u there correspond an infinite number of values of #, all these values of s give the same value to f(z) on account of its periodicity. Moreover, if U Q is a particular value of u, and s? any corresponding value of , that determination of 3 which approaches ZQ as u approaches U Q is an analytic function of u in the neighbor- hood of ; hence the same thing is true of <(w). We can therefore apply Laurent's theorem to this function < (w) In the ring-shaped region contained between the two circles C v C 2 this function is equal to the sum of. a series of the following form : HI, 66] ELLIPTIC FUNCTIONS 147 Eeturning to the variable z, we conclude from tins that in the in- terior of the cross strip consideied above the periodic function f(z) is equal to the sum of the series If the function/^) is analytic in the whole plane, we can suppose that the two straight lines LL\ J/J/', which bound the stiip, recede indefinitely in opposite directions Eveiy periodic integral function ^s therefore developable vn, a series of positive and negative poicers of convergent for every finite value of z 66 Impossibility of a single-valued analytic function with three periods. By a famous theorem due to Jacobi, a single-valued analytic function cannot have more than two independent periods To prove this we shall show that a single- valued analytic function cannot have three independent periods * Let us first prove the following lemma Let a, 6, c be any three real or complex quantities, and m, n, p three arbi- trary integers, positive or negative, of which one at least is different from zero. If we give to the integers m, w, p all systems of possible values, except m = n = p = 0, the lower limit of \ ma + nb -{- pc \ is equal to zero Consider the set (E) of points of the plane which represent quantities of the form ma + rib -f pc If two points corresponding to two different systems of integers coincide, we have, for example, ma + rib + pc = m t a + nfi + p^ and therefore (m - m x ) a + (n - njb + (p-pjc = 0, where at least one of the numbers m m x , n 74, p p x is not zero In this case the truth of the lemma is evident. If all the points of the set (E) are dis- tinct, let 2 8 be the lower limit of [ma + nb + pc \ , this number 2 S is also the lower limit of the distance between any two points whatever of the set (E) In fact, the distance between the two points ma -f nb -f pc and m 1 a + n^ 4- p^ is equal to | (m m x ) a + (n n t ) b + (p pj c |. We are going to show that we are led to an absurd conclusion by supposing $ > 0. Let N be a positive integer , let us give to each of the integers m, n, p one of the values of the sequence N, (N 1), , 0, - - , N 1, N, and let us combine these values of m, n, p, m all possible manners. We obtain thus (2 N+ I) 8 points of the set (^), and these points are all distinct by hypothesis Let us suppose |a|^|6|^|c|; then the distance from the origin to any one of the points of (E) just selected is at most equal to 3 N\ a \ These points there- fore lie in the interior of a circle O of radius BN\a\ about the origin as center or on the circle itself. If from each of these points as center we describe a * Three periods a, &, c are said to be dependent if there exist three integers m, n, p (not all zero) for which ma + rib +pc~Q TRANS 148 SINGLE-VALUED ANALYTIC FUNCTIONS [ill, 66 circle of ladms 5, all these circles will be mtenor to a cncle O x of radius equal to 3JY|a| + 5 about the origin as centei, and no two of them will overlap, since the distance between the centers of two of them cannot be smaller than 2 S The sum of the areas of all these small cucles is therefore less than the area of the cncle <?-,, and we have or (3JZV+1)*-! The right-hand side approaches zero as JST becomes infinite , hence this in- equality cannot be satisfied foi all values of N by any positive number 5 Consequently the lower limit of \ma + rib + pc \ cannot be a positive number , hence that lower limit is zeio, and the truth of the lemma is established We see, then, that when no systems of integers m, n, p (except m = n = p = 0) exist such that ma -f nb + pc = 0, we can always find integral values for these numbers such that \ma + rib + pc\ will be less than an arbitrary positive num- ber e In this case a single-valued analytic f unction f(z) cannot have the three independent periods a, 6, c For, let z be an ordinary point for/(z), and let us describe a circle o radius e about the point Z Q as centei, where e is so small that the equation f(z) =/( ) has no other root than z = z inside of this circle ( 40) If a, &, c are the periods of /(z), it is cleai that ma + nb 4- pc is also a period for all values of the integers m, n, p , hence we have /(* + met + rib + pc) =/(z ) If we choose m, n, p in such a manner that | ma -f rib + pc \ is less than , the equation f(z) =/(z ) would have a root z 1 different from 2 , where \z z \<e, which is impossible. When there exists between a, 6, c a relation of the form (20) 7na + rib+pc = Q, without all the numbers m, n 7 p being zero, a single-valued analytic function f(z) may have the periods a, 6, c, but these periods reduce to two periods or to a single period. We may suppose that the three integers have no common divisor other than unity Let JD be the greatest common divisor of the two numbers m, n , m = Dm', % = Dn' Since the two numbers m', n' are prime to each other, we can find two other integers m", n" such that m'n" m'V = 1 Let us put m'a + rib = a', m"a + n"b = &', then we shall have, conversely, a = n"af w/o', b = m'6 7 m"af If a and 6 are periods of /(), a" and b' are also, and conversely Hence we can replace the system of two periods a and b by the system of two periods a' and V The re- lation (20) becomes Daf -f pc = , D and p being prime to each other, let us take two other integers & and p' such that Dp" ~ D'p = 1, and let us put I/of -t- #'c = c' We obtain from the preceding relations of = pc', c = DC', whence it Is obvious that the three periods a, 6, c are linear combinations of the two penods 6 7 and c' As a corollary of the preceding lemma we see that if a and ft are two real quantities and m, n two arbitrary integers (of which at least one is not zero), the lower limit of | ma + np | is equal to zero For if we put o = a, & = , c i, Ill, 67] ELLIPTIC FUNCTIONS 149 the absolute value of ma -f- up + pi can be less than a number e < 1 only if we have p = 0, | ma -f n| <e Piom this it follows that a single-valued analytic function f(z) cannot have two leal independent periods a. and $ If the quotient /5/a is iriational, it is possible to find tvv o numbers m and n such that [ma -f n/5 1 is less than , and it will be possible to carry thiough the reasoning just as before If the quotient $/a is rational and equal to the irreducible fraction m/n, let us choose two integeis m! and n' such that mn' m'n = 1, and let us put m'a n'p = y The number 7 is also a period, and from the two relations manp^ 0, m'an'p = 7 we derive or := 117, /S = 9717, so that a and /3 are multiples of the single period 7 More generally, a single-valued analytic function /() cannot have two independent periods a and 6 whose ratio is real, for the function f(az) would have the two real penods 1 and &/* 67. Doubly periodic functions. A doubly periodic function is a single-valued analytic function having two periods whose ratio is not real. To conform to "Weierstrass's notation, we shall indicate the independent vanable by u 9 the two periods by 2 and 2 / 3 and we shall suppose that the coefficient of i in w'/o) is positive. Let us mark in the plane the points 2 CD, 4 >, 6 <>, and the points 2 <o f , 4 w', 6 o>', ... Through the points 2 mo let us draw parallels to the FIG 25 direction 0o> r , and through the points 2mV parallels to the direc- tion 0(D. The plane is divided in this manner into a net of congruent parallelograms (Eig. 25) Let f(u) be a single-valued analytic function with the two periods 2o>, 2 a/; from the two relations f(u + 2 o>) =/(V), f(u + 2& l )t=sf(u) we deduce at once * It is now easy to prove that there exists for any periodic single-valued f unction at least one pair of periods in terms of which any other period can be expressed as an integral linear combination , such a pair is called a primitive pair of per tods TRANS. 150 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 67 f(u + 2 m<*> -f 2 ?H V) = /(ze), so that 2 ?wo> 4- 2 ?>tV is also a period for all yalues of the integeis m and m' We shall repiesent this general penod by 2w The points that lepresent the various periods are precisely the vertices of the preceding net of paiallelograins When the point u describes the parallelogram OABC whose vertices are 0, 2 <o, 2 w 4- 2 <*>', 2 o) r , the point ^t, + 2w describes the parallelogiani whose vertices are the points 2 w, 2 w + 2 o>, 2 ?0 + 2 o> -h 2 o>', 2w + 2 o>' ? and the function /(w) takes on the same value at any pair of coirespondmg points of the two parallelograms. Every parallelogram whose ver- tices are four points of the type , w + 2 , + 2 ', w + 2 w + 2 co' is called a parallelogram of periods , in general we consider the parallelogram OABC, but we could substitute any point in the plane for the oiigm. The period 2 w + 2 <o f will be designated for brevity by 2 <D"; the center of the paiallelogram OABC is the point o>", while the points <o and CD' are the middle points of the sides OA and OC Every integral doully periodic function is a constant In fact, let f(u) be a doubly periodic function , if it is integral, it is analytic in the parallelogram OABC, and the absolute value of f(ii) remains always less than a fixed number M in this parallelogram But on account of the double periodicity the value of /(w) at any point of the plane is equal to the value of f(u) at some point of the parallelogram OABC. Hence the absolute value of f(u) remains less than a fixed number M It follows by Liouville's theorem that f(u) is a constant. 68. Elliptic functions. General properties. It follows from the pre- ceding theorem that a doubly periodic function has singular points in the finite portion of the plane, unless it reduces to a constant. The term elliptic function is applied to functions which are doubly periodic and analytic except for poles In any parallelogram of penods an elliptic function has a certain number of poles , the num- ber of these poles is called the order of the function, each being counted according to its degree of multiplicity *. It should be noticed that if an elliptic function f(u) has a pole U Q on the side OC, the point U Q 4- 2 w, situated on the opposite side AB, is also a pole , but we should count only one of these poles in evaluating the number of poles contained m OABC. Similarly, if the origin is a pole, all the * It is to be understood that the parallelogram is so chosen that the order is as small as possible Otherwise, the number of poles in a parallelogram could be taken to be any multiple of this least number, since a multiple of a period is a period TRANS (See also the footnote, p 149 ) Ill, ELLIPTIC FUNCTIONS 151 veitices of the net are also poles off(u\ but we should count only one of them in each parallelogram If, foi example, we move that vertex of the net which lies at the origin to a suitable point as near as we please to the origin, the given function /(K) no longei has any poles on the boundaiy of the parallelogiam When we have occa- sion to integrate an elliptic function /() along the boundary of the parallelogram of periods, we shall alwa} s suppose, if it is necessary, that the parallelogram has been displaced in such a way that f(u) has no longer any poles on its boundaiy The application of the general theorems of the theoiy of analytic functions leads quite easily to the fundamental piopositions 1) The sum of the residues of an elliptic function with respect to the poles situated in a parallelogram of periods is zero Let us suppose for definiteness that f(ii) has no poles on the boundary OABCO The sum of the residues with respect to the poles situated within the boundary is equal to the integral being taken along OABCO But this integral is zero, for the sum of the integrals taken along two opposite sides of the paial- lelogram is zero Thus we have c J(OA) and if we substitute u -f- 2 o>' for u in the last integial, we have r f(u)du= c f(u+2u r )du=* f/(w)<fo=- c /oo */() *A ^2toi J(OA) Similarly, the sum of the integrals along AB and along CO is zero In fact, this piopeity is almost self-evident from the figure (Fig 26). For let us consider two corresponding elements of the two inte- grals along OA and along BC* At the points m and m 1 the values of f(u) are the same, while the values of du have opposite signs The preceding theorem proves that an elliptic func- tion f(u) cannot have only a single pole of the first order in a parallelogram of periods. An elliptic function is at least of the second order, FIG. 152 SINGLE-VALUED ANALYTIC FUNCTIONS [HI, 68 2) The number of zeros of an elliptic function m a parallelogram of periods is equal to the order of that function (each of the zeros being counted according to its degree of multiplicity) Let/(?/) be an elliptic function, the quotient /(w)//() == </>(ii) is also an elliptic function, and the sum of the residues of <f> (it) in a par- allelogram is equal to the number of zeros of f(if) diminished by the numbei of the poles ( 48) Applying the pieceding theoiem to the function <(?/), we see the truth of the proposition ]ust stated In gen- eial, the numbei of roots of the equation f(it) = C in a paiallelogram of periods is equal to the oidei of the function, for the function /(?/) C has the same poles as /()> whatever may be the constant 0. 3) The difference between the sum of the zeros and the sum of the poles of an elliptic function in a parallelogram of periods is equal to a period. Consider the integral 1 r o I 2-iriJ , u ^7~i du /(M) along the boundary of the parallelogram OAEC This integral is equal, as we have already seen ( 48), to the sum of the zeios of /() within the boundaiy, diminished by the sum of the poles of /(%) within the same boundary. Let us evaluate the sum of the integrals lesulting from the two opposite sides OA and EC r Jo o If we substitute u + 2 <a' for u in the last integral, this sum is equal to or, on account of the periodicity of f(u), to The integral f Jo is equal to the variation of Log[/(w)] when u describes the side OA\ but since /(w) returns to its initial value, the variation of Log [/(%)] is equal to 2 m 2 iri 9 where m 2 is an integer The sum of the inte- grals along the opposite sides OA and EC is therefore equal to Ill, 68] ELLIPTIC FUNCTIONS 153 (4 M 2 7rio> r )/2 iri = 2???X Similarly, the sum of the integrals along AB and along CO is of the form 2 m^ The diffeience considered above is theiefoie equal to 2 m^ + 2 w 2 co' , that is, to a period By a smiilai argument it can be shown that the proposition is also applicable to the loots of the equation /(w) = C, contained in a parallelogram of penods, foi any value of the constant C. 4) Between any two elliptic functions with the same periods there exists an algebraic relation Let /(?/), / x (w) be two elliptic functions with the same periods 2o>, 2co'. In a parallelogram of periods let us take the points a l9 a^ , a m which aie poles for either of the two functions f(u), / x (w) or foi both of them, let^t, be the higher oider of multi- plicity of the point a t with lespect to the two functions, and let A& x + Aj + + A& m = N. ^ow let F(v, y) be a polynomial of degree n with constant coefficients If we replace x and y by f(it) and f^u*), respectively, in this polynomial, theie will lesult a new elliptic func- tion $ (it) which can have no othei poles than the points a l9 # , - , a m and those which are deducible from them by the addition of a period. In order that this function <>(?*) may reduce to a constant, it is necessaiy and sufficient that the principal paits disappeai in the neighborhood of each of the points a v a 3 , - -, a m . Now the point a l is a pole for < (u) of an order at most equal to n^ Wntmg the con- ditions that all the principal parts shall be zero, we shall have then, in all, at most n Oi + A*a + ---- M) = Nn linear homogeneous equations between the coefficients of the poly- nomial F(x, y) in which the constant teim does not appear There are n(n + 3)/2 of these coefficients, if we choose n so large that n(n + 3)> 2Nn, or n + 3 > 2 X, we obtain a system of linear homogeneous equations in which the number of unknowns is greater than that of the equations Such equations have always a system of solutions not all zero. If F(x 9 y) is a polynomial determined by these equations, the elliptic functions f(u\ f^u) satisfy the algebraic relation where C denotes a constant Notes. Before leaving these general theorems, let us make some further observations which we shall need later A single-valued analytic function /(?) is said to be even if we have /( ) =/(w) , it is said to be odd if we have /( w) =/(). 154 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 68 The derivative of an even function is an odd function, and the derivative of an odd function is an even function In geneial, the derivatives of even older of an even function are themselves even functions, and the derivatives of odd older are odd functions. On the contrary, the deiivatives of even order of an odd function are odd functions, and the deiivatives of odd order aie even functions Let/(w) he an odd elliptic function, if w is a half-peiiod, we must have at the same time /<V)=-/(- w) and/(w)=/(- w), since % = iff + 2 w K 1S necessaiy, then, that f(w) shall be zero or infinite, that is, that w must be a zero 01 a pole for f(u) The order of multiplicity of the zeio 01 of the pole is necessarily odd, if w weie a zeio of even order 2 for/(), the derivative /< 2n) (>), which is odd, would be analytic and different from zero for u =? w If w were a pole of even order for/(w), it would be a zero of even order foi 1/f (u). Hence we may say that every half-period is a zero or a, pole of an odd order for any odd elliptic function. If an even elliptic function /(u) has a half-period w for a pole or for a zero, the order of multiplicity of the pole or of the zero is an even number If, for example, w were a zero of odd order 2 n + 1, it would be a zero of even order for the derivative /'(*), which is an odd function The proof is exactly similar for poles Since twice a period is also a period, all that we have just said about half-periods applies also to the periods themselves. 69. The function p(w). We have already seen that every elliptic function has at least two simple poles, or one pole of the second order, in a parallelogram of periods In Jacobi's notation we take func- tions having two simple poles for our elements ; in Weierstrass's notation, on the contrary, we take for our element an elliptic func- tion, having a single pole of the second order in a paiallelogram Since the residue must be zero, the principal part in the neighbor- hood of the pole a must be of the form. A/(u df. In older to make the problem completely definite, it suffices to take A=l and to suppose that the poles of the function are the origin u = and all the vertices of the network 2w = 2mo> + 2mV. We are thus led first to solve the following problem : To form an elliptic function having as poles of the second order all the points 2 w = 2 m<o + 2 m'w', where m and m! are any two integers whatever, and having no other poles, so that the principal part in the neighborhood of the point 2 w shall be l/(w Ill, 69] ELLIPTIC FUNCTIONS 155 Before applying to this problem the geneial method of 62, we shall first prove that the double senes where m and m 1 take on all the integral values fiom oo to 4- oo (the combination m = m' = being excepted), is convergent, provided that the exponent p is a positive number greater than 2. Consider the triangle having the thiee points u = 0, u = wo>, u = ma 4- mV for its vertices , the lengths of the three sides of the triangle are respec- tively |mo>|, |mV|, \m<o + wV| We have, then, the relation where is the angle between the two directions 0&, 0o> f (0 < B < TT) For bievity let ||= a, |o>'| = 5, and let us suppose a ^. The pie- ceding relation can then be written in the form |mo> 4 mV j 3 = ra 2 a 2 -f w' 2 Z 2 2 mm'ab cos <>, where the angle is equal to if ^ Tr/2, and to TT ^ if > Tr/2 The angle cannot be zero, since the three points 0, CD, <o f are not 111 a straight line, and we have ^ cos < 1. "We have, then, also |m<o + m r co r | 2 = (1 cos <>) (w 2 a 2 + m 12 ^ 2 ) 4- cos (ma m f ^) 2 , and consequently | mo 4- m'<o f | 2 ^ (1 cos ) (m 2 2 4 w' 2 ^ 2 ) S (1 cos ) a 2 (w 2 -f m 12 ). From this it follows that the terms of the senes (21) are respectively less than or equal to those of the senes S'l/(#t 2 -f m 18 )'* 72 multiplied by a constant factor, and we know that the last series is convergent if the exponent p/2 is greater than unity (I, 172) Hence the senes (21) is convergent if we put /<& = 3 01 p> = 4. According to a result derived in 62, the series represents a function that is analytic except for poles, and that has the same poles, with the same principal parts, as the elliptic function sought. We shall show that this function <f> (u) has precisely the two periods 2 o> and 2 a/ Consider first the series where 2 w = 2 m<# 4- 2 m f <D f 3 the summation being extended to all the -integral xstLiies of m and m', except the combinations m = m f = 156 SINGLE-VALUED ANALYTIC FUNCTIONS [in, 69 and m = 1, m T = 0. This series is absolutely convergent, for it lesults from the series $(ii) when we substitute 2<o foi u and omit two teims. It is easily seen that the sum of this series is zero by considering it as a double series and evaluating separately each of the rows of the rectangular double array Subtracting this series from <(V), we can then wiite i <o) 2 J' the combinations (w=w r = 0), (m = 1, m' = 0) being always excluded from the summation Let us now change u to u 2 <o , then we have the combination m = 1, ra f = being the only one excluded fiom the summation. But the right-hand side of this equality is identical with (w). This function has therefoie the period 2 o>, and in like manner we can prove that it has the period 2 o>' This is the func- tion which Weierstrass represents by the notation p(w), and which is thus defined by the equation (22) P() = + If we put u = in the difference p (u) 1/w 2 , all the terms of the double sum are zero, and that difference is itself zero The function p(^) possesses, then, the following properties 1) It is doubly periodic and has for poles all the points 2 w and only those. 2) The principal part in the neighborhood of the origin is 1/M 2 3) The difference p(w) 1/w 2 is zero for u = 0. These properties characterize the function p (u) In fact, any analy- tic function f(u) possessing the first two properties differs from p (u) only by a constant, since the difference is a doubly periodic func- tion without any poles. If we have also f(u) 1/V = for u = 0, f(u) p(u) is also zero for u = ? we have, therefore, /(^) = p(w). The function p( w) evidently possesses these three properties; we have, then, p( u)= p(w), and the function p(-w) is even, which is also easily seen from the formula (22) Let us consider the period of p (u) whose absolute value is smallest, and let 8 be its absolute value. Within the circle C s with the radius B 3 described about the origin as center, the difference p(u) 1/u? is Ill, 69] ELLIPTIC FUNCTIONS 157 analytic and can be developed in positive powers of u The general term of the series (22), developed in poweis of u, gives T + - 5 4 w? 2 (2 zr) 3 ^ (2 w) 4 ^ ^ (2 w and it is easy to prove that the function. 5 u 16|wf u M dominates this series in a circle of radius 3/2, and, a fortiori, the expression obtained from it by replacing 1 u/\w\ by 1 2-w/S dominates the series Since the senes S'l/jwj 8 is convergent, we have the right to add the resulting senes term by term (9). The coefficients of the odd powers of u are zero, for the terms resulting from periods symmetrical with respect to the origin cancel, and we can write the development of p (u) in the form (23) p(u)=^ + %u* + c z i<* + - + c x u**-* + --, where (24) Whereas the formula (22) is applicable to the whole plane, the new development (23) is valid only in the interior of the circle C B hav- ing its center at the origin and passing through the nearest vertex of the periodic network. The derivative p'(w) is itself an elliptic function having all the points 2w foi poles of the third ordei. It is represented in the whole plane by the series (25) pW= _2_ In general, the nth derivative p (n) (?/) is an elliptic function having all the points 2 w for poles of order n + 2, and it is represented by the series (26) We leave to the reader the verification of the correctness of these developments, which does not present any difficulty in view of the properties established above ( 39 and 61) 158 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 70 70. The algebraic relation between p(i/) and p'(i/). By the general theorem of 68 theie exists an algebraic relation between p(w) and p'(w). It is easily obtained as follows In the neighborhood of the origin we have, from the foiinula (23), where the terms of the senes not written are zero for u = The difference ?*() - 4 p*( w ) has tn ^ refore the 011 S m as a P ole of tlie second order, and in the neighborhood of this point we have where the terms not written are zeio for u = Hence the elliptic function 20 cjp(u) 28 c z has the same poles, with the same principal parts, as the elliptic function p' 2 - 4p 8 , and their difference is zero when u = These two elliptic functions are therefore identical, and we have the desired idation, which we shall write in the form (27) [p'(f O] 1 = 4 pF() - ff j> (u) - where The relation (27) is fundamental in the theory of elliptic func- tions ; the quantities g 2 and g z are called the invariants AH the coefficients o x of the development (23) are polynomials in terms of the invariants g 2 and g 9 In fact, taking the derivative of the relation (27) and dividing the result by 2 P'(M), we derive the formula (28) p On the other hand, we hare in the neighborhood of the origin "() = 1 + 2e a + 12e^+ . . -f-(2X- 2)(2X - 8)^*-*+ m,7l] ELLIPTIC FUNCTIONS 159 Replacing p(w) and p"(w) by their developments in the relation (28), and remembering that (28) is satisfied identically, we obtain the recurrent relation Cti *? Q /\ Q\T V = ^, O, - *, (A. 6)\, which enables us to calculate step by step all the coefficients C A in terms of c 2 and <? 3 , and consequently in terms of g^ and g z , we find thus ,, = ____ * 2 4 3 5 2 ' 5 ~2*.5.7 11 ' This computation brings out the remarkable algebiaic fact that all the sums S'l/(2 w) Zn are expressible as polynomials in terms of the first two We know a pnori the roots of p'(w). This function, being of the thud cider, has thiee roots in each parallelogram of periods Since it is odd, it has the roots u = o, u = /, u = &" = o> + ' ( 68, notes). By (27) the roots of the equation 4 p 8 <?jp ^ 3 = are precisely the values of p () for w = o>, CD', co". These three roots are ordinarily represented by e v e^ e^ : *i = P ( to ). a = P C* 1 )* 6 s = P ( w ")- These three roots are all different ; for if we had, for example, ^ = 2 , the equation p(w)= e 1 would have two double roots <o and o> f in the interior of a parallelogram of periods, which is impossible, since p(w) is of the second order Moreover, we have and between the invariants g^ g s and the roots e v e 2 , e 8 we have the relations e i + 6 * + \ = > e i e 2 + e i e s + W* = - 'f ' e iV* = f The discriminant (*^ 27 <7l)/16 is necessarily different from zero. 71. The function (u). If we integrate the function p(^) along any path whatever starting from the origin and not passing through any pole, we have the relation The series on the right represents a function which is analytic except for poles, having all the points u = 2w, except u = 0, for 160 SINGLE-VALUED ANALYTIC FUNCTIONS [m, 71 poles of the fiist order Changing the sign and adding the frac- tion 1/u, we shall put (29) The preceding relation can be written (30) rT ' % 11 * '" ^ ' 1 and, taking the derivatives of the two sides, we find It is easily seen from either one of these formulae that the function (u) is odd. In the neighborhood of the origin we have by (23) and (30), The function (u) cannot have the periods 2 <o and 2 a/, for it would have only one pole of the first order in a parallelogram of periods. But since the two functions (u 4- 2 w) and (u) have the same deriva- tive p (u), these two functions differ only by a constant , hence the function (u) increases by a constant quantity when the argument u increases by a period It is easy to obtain an expression for this con- stant. Let us wiite, for greater clearness, the formula (30) in the form Changing it, to u -f- 2 u> and subtracting the two f ormulse, we find C( + 2)-C()= r "p(v)dv. Jit We shall put xiu+20) rM + 2i' 217= I p(v)dv, 2i/:= / p(v)dv. Then 17 and ^ r are constants independent of the lower limit u and of the path of integration This last point is evident a priori, since all the residues of p(v) are zero. The function (u) satisfies, then, the two relations f( + 2 ) = {(*) + 2 % {(u + 2 */) = f If we put in these formulae w = o> and w = o r respectively, we find 17 = (*>), ,/ =(0,'). Ill, 71] ELLIPTIC FUNCTION'S 161 There exists a very simple relation between the four quantities o>, a/, iq, iff. To establish it we have only to evaluate in two ways the integral J(ii)du, taken along the parallelogram whose vertices are U Q , z* -f 2 <o, + 2 (o -f 2 o', w -j- 2 a/, "We shall suppose that () has no poles on the boundary, and that the coefficient of i in o>'/<o is positive, so that the veitiees will be encountered in the order in which they aie written when the boundary of the parallelogiam is described in the positive sense There is a single pole of (it) in the interior of this boundary, with a residue equal to -f- 1 5 hence the integral under consideration is equal to 2 TTI On the other hand, by 68 the sum of the mtegials taken along the side joining the veitiees U Q> U Q -|- 2 <D and along the opposite side is equal to the expiession / t/Uft - 4:0)77'. Similarly, the sum of the integrals coming from the other two sides is equal to 4 0/17 We have, then, /OO\ f ___ f 2H a" which is the relation mentioned above. Let us again calculate the definite integral "**(*)*>, taken along any path whatever not passing through any of the poles We have ^ f/ N &/ so that F(u) is of the form F(ii) ~2iju + K, the constant K being determined except for a multiple of 2 iri y for we can always modify the path of integration without changing the extremities in such a way as to increase the integral by any multiple whatever of To find this constant K let us calculate the definite integral along a path, very close to the segment of a straight line which joins the two points o> and . This integral is zero, for we can replace the path of integration by the rectilinear path, and the elements of the new integral cancel in pairs. But, on replacing u by o> in the expression which gives F(w), we have /vhw I ^(v)^ = - J 6 162 SINGLE-VALUED ANALYTIC FUNCTIONS [in, 71 and since we have also we can take K = 2 yu iri Hence, without making any supposition as to the path of integiation, we have, in general, (33) C" Ju where m is an integer, and we have an analogous formula for the integral f^ +2 ^(v)du 72. The function CT(M). Integrating the function (w) l/ along any path starting from the origin and not passing through any pole, we have and consequently (34) .jrDw-g*- Ml The integral function on the right is the simplest of the integral functions which have all the periods 2 w for simple roots , it is the function <r(z) (35) ,()- .n--. The equality (34) can be written (34') o-() = whence, taking the logarithmic derivative of both sides, we obtain The function <r(^), being an integral function, cannot be doubly periodic. When its aigument increases by a period, it is multiplied by an exponential factor, which can be determined as follows : From the formula (34*) we have cr(u) U This factor was calculated in 71, whence we find (37) cr(u Ill, 73] ELLIPTIC FUNCTIONS 163 It is easy to establish in a similar manner the relation (38) o-O + 2 ') = - eW+"><r(ii). From either of the formulae (35) or (34') it follows that <r(w) is an odd function If we expand this function <r(w) in powers of u, the expansion obtained will be valid for the whole plane It is easy to show that all the coefficients are polynomials in g Q and g y For we have 3.4" 5.6" 2X(2X-1) <r(^)=we"" r * U *"" r ^ we ~ ". We see that there is no term in if and that any coefficient is a polynomial in the c A 3 s and theiefoie in the invariants #, and ^ 8 , the first five terms are as follows . " 2 4 35 2*357 2 9 3 2 5.7 2 7 .3 2 5 2 .7.11 The three functions p(z), ()? *00 are ^he essential elements of the theory of elliptic functions The first two can be derived from or(w) by means of the two relations (u) = <r'()/<r(w), p(u) = '(M) 73. General expressions for elliptic functions. Every elliptic function f(u) can be expiessed in terms of the single function <r(z), or again in terms of the function (u) and of its derivatives, or finally in terms of the two functions p(w) and p'(z) We shall present con- cisely the three methods. Method 1. Expression of f(u) in terms of the function <r(u). Let a i> a Q> ' '9 a n ^ e ^ ne zeros ^ tne function f(u) in a parallelogram of periods, and l lt 1 9 - , Z> n the poles o/(w) in the same parallelogram, each of the zeros and each of the poles being counted as often as is required by its degree of multiplicity. Between these zeros and poles we have the relation (40) r l + a s| +.. +a n = b l + lt+- where 2 O is a period Let us now consider the function This function has the same poles and the same zeros as the function f(u), for the only zeros of the factor ar(u a t ) are u = a t and the 164 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 73 values of u which differ from a % only by a period On the other hand, this function <(M) is doubly penodic, for if we change u to u -f- 2 o>, for example, the relation (37) shows that the numerator and the denominator of <f>(u) are multiplied lespectively by the two factors and these two factors are equal, by (40) Similaily, we find that <^(u -h 2o>') = <f>(u). The quotient f(u)/<]> (11) is therefore a doubly periodic function of u having no infinite values , that is, it is a constant, and we can write tl \ r i /W- c ^ _ b ^ u _ ^ ^ _ ^ To determine the constant C it is sufficient to give to the vaiiable u any value which is neither a pole nor a zero More generally, to express an elliptic function /() in terms of the function <r(u), when we know its poles and its zeios, it will suf- fice to choose n zeros (a^, a^ , <) and n poles (l^ b& - , b) in such a way that Sd^ = S^^ and that each root of f(u) can be obtained by adding a period to one of the quantities af, and each pole by adding a period to one of the quantities % These poles and zeros may be situated in any way in the plane, provided the preceding conditions are satisfied Method 2. Expression off(u) in terms of the function and of its derivatives. Let us consider k poles a 1? a 2 , -, a L of the function/(w) such that every other pole is obtained by adding a period to one of them We could take, for example, the poles lying in the same parallelogram, but that is not necessary. Let (u a t ) w * be the principal part of f(u) in the neighborhood of the point a t . The difference is an analytic function in the whole plane. Moreover, it is a doubly periodic function, for when we change u to u 4- 2 o>, this function is increased by 2^A^, which is zero, since S^i represents the sum HI, 73] ELLIPTIC FUNCTIONS 165 of the residues in a parallelogiam. That difference is theiefore a constant, and we have ;= C + A>t(u - a.) - Jj,r t H - a,) - - - (42) -=I L -K-i^^ji^-^ 0] The preceding formula is due to Hermite. In order to apply it we must know the poles of the elliptic function f(ii) and the corre- sponding principal parts Just as formula (41) is the analogon of the foimula which expresses a lational function as a quotient of two polynomials decomposed into then lineai factors, the formula (42) is the analogon of the formula for the decomposition of a rational fraction into simple elements. Here the function (u a) plays the part of the simple element. Method 3 Expression off(u) in terms ofp(i() and ofp'(u) Let us consider first an even elliptic function f(u) The zeros of this function whith 0,1 e not pet fads, are symmetric in pairs We can theiefore find n zeios (a j9 a 2 , , a n ) such that all the zeros except the penods are included in the expressions We shall take, for example, the parallelogram whose vertices aie <o + <o f , a/ o>, a) <>', co <o r and the zeios in this parallelogram lying on the same side of a straight line passing through the origin, carefully excluding half the boundary in a suitable manner. If a zero a t is not a half-period, it will be made to appear in the sequence a l9 a 2) , a n as often as there are units in its degree of multiplicity. If the zero a v for example, is a half-period, it will be a zero of even order 2 r ( 68, notes) We shall make this zero appear only r times in the sequence a l9 a 2 , , a n With this understanding, the product has the same zeros, with the same orders, as /(*), excepting the case of /(O) = Similarly, we shall form another product, having the poles of f(u) for its zeros and with the same orders, again not considering the end points of any period. Let us put CP () ~ P Ml . 166 SINGLE-VALUED ANALYTIC FUNCTIONS [m, 73 the quotient /00/<K'0 1S an elliptic function which has a finite value different ft oni zem for eveiy value of u which is not a penocl This elliptic function i educes to a constant, for it could only have periods for poles , and if it did, its leciproeal would not have any poles We have, then, ,/, A _ r EP 00 - P (i)] [P fr ) ~ P K)] CP 00 - P (<Q] . /( } "" [POO- PCOTPOO- P&)] [POO- P&.)] If / 1 (?/) is an odd elliptic function, ,/iOO/P'OO 1S an ev ^ n function, and therefore this quotient is a lational function of p (z) Finally, any elliptic function F(tt) is the sum of an even function and an odd function . Applying the preceding results, we see that every elliptic function can be expressed in the form (43) J P()= *[>()] + ?'()*,!>()], where -R and 72 1 are rational functions 74 Addition formulae. The addition formula for the function sin x enables us to express sin (a 4- Z>) in terms of the values of that func- tion and of its derivative for x = a and x = I There exists an analogous formula for the function p(w), except that the expression for p(u -f- #) in terms of p(^), p(v), p'(w), p'(*0 is somewhat more complicated on account of the presence of a denominator Let us first apply the general formula (41), in which the function cr(u) appears, to the elliptic function p(w)""P(^) ^ e see a ^ once that cr(u -I- v) <r(u v)/cr\ii) is an elliptic function with the same zeros and the same poles as p (w) p (y) We have, then, / \ /\ ^ < p()_p( B )-C in order to determine the constant C it suffices to multiply the two sides by &*(u) and to let u approach zero We thus find the relation 1 = Ca*(v), whence we derive ,... , , ,. <r(u + - (44) PW P() = -- _j ^ / <r\/<r\/ 2 If we take the logarithmic derivative on both sides, regarding v as a constant and u as the independent variable, we find Ill, 74] ELLIPTIC FUNCTIONS 167 or, interchanging u and v in this result, Finally, adding these two results, we obtain the relation which constitutes the addition foimula for the function (u). Differentiating the two sides with respect to u, we should obtain the expression for p (u + v) , the right-hand side would contain the second derivative p"(w), which would have to be replaced by 6 p 3 (w) gJ2 This calculation is somewhat long, and we can obtain the result in a more elegant way by proving first the relation (46) p(u + v) + p(tO + POO = [C(* + *)- COO- COOT- Let us always regard u as the independent variable , the two sides are elliptic functions having for poles of the second order u = 0, u = v, and all the points deducible from them by the addition of a period In the neighborhood of the origin we have CO* + *)- COO- C00= COO+ POO+ ---- COO- COO =- ^ + *C'00+ *+ and consequently - 2 + . . -. The principal part is l/% 2 , as also for the left-hand side. Let us compare similarly the principal parts in the neighborhood of the pole u = v Putting u = v -f A, we have - C(* - ) - COOT- - 2 C'W + - - The principal part of the right-hand side of (46) m the neighbor- hood of the point u = v is, then, /(u + vf, just as for the left- hand side Hence the difference between the two sides of (46) is a constant To find this constant, let us compare, for instance, the developments in the neighborhood of the origin. We have in this neighborhood p( + v) + p(i*) 4- p(0 = 168 SINGLE- VALUED ANALYTIC FUNCTIONS [III, 74 Comparing this development with that of [(w -f r) (M) (*>)] 2 , we see that the difteience is zeio foi it = The i elation (46) is there- fore established Combining the two equalities (45) and (46) , we obtain the addition foinmla foi the function p(w) 75. Integration of elliptic functions. Hermite's decomposition for- mula (42) lends itself immediately to the integration of an elliptic function Applying it, we find We see that the integral of an elliptic function is expressible in terms of the same transcendentals or, , p as the functions themselves, but the function <r(u) may appear in the result as the argument of a logarithm In ordei that the integral of an elliptic function may be itself an elliptic function, it is necessary first that the integral shall not present any logarithmic critical points , that is, all the residues A ( f must be zero If this is so, the integral is a function analytic except foi poles In order that it be elliptic, it will suffice that it is not changed by the addition of a period to u, that is, that 2?2>-lJ> = 0, 2CV - 2^2^= 0; whence we derive C = 0, ^A ( f = 0. If these conditions are satisfied, the integral will appeal in the form indicated by Hermite's theorem. "When the elliptic function which is to be integrated is expressed in terms of p(u) and p'(w) it is often advantageous to start from that form instead of employing the general method Suppose that we wish to integrate the elliptic function R [p (w)] + p' (u) R l [p (u)], R and R l being rational functions We have only to notice in regard to the integral fR l [p (u)] p'(u) du that the change of variable p (u) = t reduces it to the integial of a rational function. As for the integral fR [p (u)] du, we could reduce it to a certain number of type forms by means of rational operations combined with suitably chosen mtegiations by parts ; but it turns out that this would amount to making in another form the same reductions that were made in Volume I ( 105, 2d ed. , 110, 1st ed.). For, if we make the change of variable p (u) = t , which gives <#, or dtt dt m,fc75] ELLIPTIC FUNCTIONS 169 the mtegral/JR [p (u)] du takes the form E(t)dt We have seen how this integral decomposes into a rational function of t and of the radical V4 t s g^t g^ a sum of a certain number of integrals of the f oim ft n dt/ V4 t s g 2 t </ 3 , and finally a certain number of integrals of the form dt j wheie P() is a polynomial prime to its derivative and also to 4 3 g z t g^ and where Q (t) is a polynomial prime to P (t) and of lower degree than P (t) Returning to the variable u, we see that the mtegial fR[p(u)]du is equal to a rational function of p(ic) and p'(tt), plus a certain number o integrals such asJ"[p(u)] n <Zit and a certain number of other integrals of the form r J P &>()] ' and this reduction can be accomplished by rational operations (multiplications and divisions of polynomials) combined with certain integrations by parts. We can easily obtain a recuirent formula for the calculation of the integrals I n = J*[p (u)1 n du If, m the i elation {[P ()]- WM)} = (-!) [P M]- 2 P' 2 (u) + [p (tt)]- ip"(), we replace p /2 (w) and p /7 (w) by 4p 3 (w) gr 2 p(w) gr s and 6p 2 (w) gr 2 /2 respectively, there results, after arranging with respect to and from this we derive, by integrating the two sides, / i\ (50) [p(u)'] n ~- 1 p'(u) = (4ra + 2) I n + i in ir 2 J n _i (?i l)gr 3 I B _ 2 By putting successively % = 1, 2, 3, - m this formula, all the integrals I n can be calculated successively from the first two, J = w, Z 1 = f (u) To reduce further the integrals of the form (49), it will be necessary to know the roots of the polynomial P(t). If we know these roots, we can reduce the calculation to that of a certain number of integrals of the form du where p(v) is different from ^, <? 2 , e 8 , since the polynomial P(t) is "prime to 4 $8 _ gj _ g The value of u is therefore not a half-period, and p'(v) is not zero. The formula established in 74, then gives (51) C - ^L_^ = ^ v ' Jp(w)-p() p'() 170 SINGLE-VALUED ANALYTIC FUNCTIONS [ill, 76 76. The function B The series by means of which we have defined the func- tions p(w), (u), ff(u) do not easily lend themselves to numerical computation, including even the power series development of <r(u), which is valid for the whole plane The founders of the theory of elliptic functions, Abel and Jacobi, had introduced another remarkable transcendental, which had previously been encountered by Fourier in his work on the theory of heat, and which can be developed in a very rapidly convergent series , it is called the 6 function We shall establish briefly the principal properties of this function, and show how the Weierstrass <r (u) function can be easily deduced from it Let r = r + si be a complex quantity in which the coefficient a of i is positive If v denotes a complex variable, the function 6 (v) is defined by the series - +w / ]\a (52) 6 (V) =~S (- l)gfV* + *J e fl + I)w q _ girtr^ 1 ^* 00 . which may be regarded as a Laurent series in which & w has been substituted for z This series is absolutely convergent, for the absolute value U n of the general term is given by if u = a + j8z , hence V U n approaches zero when n becomes infinite through positive values, and the same is true of Vt71_ n It follows that the function 8 (o) is an integral transcendental function of the variable u It is also an odd function, for if we unite the terms of the series which correspond to the values n and n 1 of the index (where n varies from to +00), the development (52) can be replaced by the following formula (53) 0(v) = 2 V ( l) n gv + 2) sin(2n+l)7n>, o which shows that we have Q (-fl)=- 0(v), 0(0) = When v is increased by unity, the general term of the series (52) is multi- plied by e (2+ !>** = - 1 We have, then, 6 (v + 1) = 8 (v) If we change 1? to v + T, no simple relation between the two series is immediately seen , but if we write r) = ] (- l)n 00 and then change n to n 1 in this series, the general term of the new series is equal to the general term of the series (52) multiplied by q- 1 e-*" w . Hence the function Q (v) satisfies the two relations (54) 0(t> + l)=-0(t>), 0(t> + r)=-g-i er 2inv0( l ,) Since the origin is a root of 6 (t>), these relations show that Q (v) has for zeros all the points m^ + m 2 T, where m^ and m 2 are arbitrary positive or negative integers III, 76] ELLIPTIC FUXCTIOXS 171 These are the only roots of the equation (t) = For, let us consider a parallelogram whose vertices are the four points v , v -f 1, V Q -f 1 + T, t? + r, the fhst veitex V Q being taken m such a -way that no loot of 0(e) lies on the boundaiy We shall show that the equation 0(v) = has a single root in this parallelogiam For this purpose it is sufficient to calculate the integral along its boundary in the positive sense By the hypothesis made upon r, we encounter the vertices in the order in which they aie wntten From the relations (54) we derive .. s 0<B) The fiist of these relations shows that at the coiiespondmg points n and n' (Fig 27) of the sides AD, BC, the function 0' ()/#() takes on the same value Since these two sides are described in contrary senses, the sum of the eor- responding integrals is zero On the contrary, if we take two corresponding points m mf on the sides J.J3, DC, the value of 0'0?)/0(o) at the point m' is equal to the value of the same function at the point m, diminished by %m. The sum of the two mtegials coming from FIG 27 these two sides is therefore equal to /(CD) ~~ ^ fl^^i fck at is i to 2 m As there is evidently one and only one point in the parallelogram A BCD which is represented by a quantity of the form m 1 4- m s r, it follows that the function 6 (v) has no other roots than those found above. Summing up, the function B (v) is an odd integral function , it has all the points m l + m z r for simple zeros , it has no other zeros , and it satisfies the relations (54) Let now 2 w, 2 ' be two periods such that the coefficient of i in wVo> is positive In 9 (a>) let us replace the variable V by ie/2 w and r by w'/w, and let < (w) be the function (55) *(> = Then $ (u) is an odd integral function having all the periods 2 w = 2 mw -f for zeros of the first order, and the relations (54) are replaced by the following (56) These properties are very nearly those of the function <r (u) In order to re- duce it to ff (u), it suffices to multiply # (w) by an exponential factor. Let us put (57) Vfa) = ^^%(M), where 57 is the function of w and w' defined as m 71. This new function ^ (u) is an odd integral function having the same zeros as (M) The first of the 172 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 76 relations (56) becomes (58) f(ii + a)=-^5= fII+9li %W=-e 1 ^ + - ) *W- We have next or, since i?t/ i?' = iri/2, (59) I (u + 2 ') = - e 2 *< + >'> ^ (M) The relations (58) and (59) aie identical with the relations established above for the function <r(u) Hence the quotient t(u)/<r(u) has the two penods 2w and 2 u', for the two teims of this latio aie multiplied by the same factor when u increases by a period Since the t\vo functions have the same zeros, this quotient is constant , moieover, the coefficient of u in each of the two develop- ments is equal to unity We have, then, a- (u) = $ (u), 01 (60) , ( and the function <r(u) is expressed in terms of the function 0, as we proposed If we give the argument v real values, the absolute value of q being less than unity, the senes (53) is rapidly convergent We shall not further elaboiate these indications, which suffice to suggest the fundamental part taken by the 6 function m the applications of elliptic functions III INVERSE FUNCTIONS CURVES OF DEFICIENCY ONE 77. Relations between the periods and the invariants. To every system of two complex numbers w, <>', whose ratio o>'/a> is not real, corresponds a completely determined elliptic function p(u), which has the two periods 2 o>, 2 a>', and which is regular for all the values of u that are not of the foim 2 mo> + 2 w V, all of which are poles of the second order The functions (u) and <r(u) 9 which are deducible fiom p(u) by one or by two integrations, respectively, are likewise determined by the system of periods (2 <o, 2 <o f ). When there is any reason for indicating the penods, we shall make use of the notation P(K|O, *>'), (w|> <> f )? <r(w|> <>') to denote the three fundamental functions. But it is to be noticed that we can replace the system (w, o> f ) by an infinite number of other systems (O, O 1 ) without changing the function p(w). For let m, m f , n, n 1 be any four positive or negative integers such that we have mn' m'n = 1 If we put we shall have, conversely, <u =s (n'Q %O') 3 o)' =3 (mO' m'O), Ill, 77] INVERSE FUNCTIONS 173 and it is clear that all the periods of the elliptic function p(*f) are combinations of the two periods 2Q, 2Q', as well as of the two periods 2 <o, 2 >' The two systems of periods (2 <D, 2 *>') and (2 O, 2 O') are said to be equivalent The function p(|Q, G') has the same periods and the same poles, with the same principal parts, as the function p(w|<o, o>'), and their difference is zeio for u = 0. They are therefore identical This fact results also fiorn the development (22), for the set of quantities 2 m<* + 2 ??zV is identical with the set of quantities 2mO-j-2w'Q'. For the same reason, we have i(w|O, O')= 0jo>, >') and <T(H|Q, O') = <r(|o>, o/) Similarly, the three functions p(w), (?<), <r(z*) are completely deter- mined by the invariants ^ 2 , ^ 3 For we have seen that the function <r (u) is represented by a power-series development all of whose coeffi- cients are polynomials in <? , g^ We have, then, (?/) = o- f (i^)/<r (M), and finally p(w) = '(w). In older to indicate the functions which coriespond to the invanants </, and g$ we shall use the notation P ( u : ff <7 8 )> > ^ ^s)? "C^ 5 &, ^)- Just here an essential question piesents itself. While it is evi- dent, from the veiy definition of the function p (?;), that to a system (o>, <o f ) corresponds an elliptic function p(w), provided the ratio o)'/<o is not real, there is nothing to prove a priori that to every system of values for the invariants g^ g z corresponds an elliptic function. "We know, indeed, that the expression g\ 27 #f must be different from zero, but it is not certain that this condition is suffi- cient The problem which must be treated here amounts in the end to solving the transcendental equations established above, (61) fc- 60 2'(2i + 2 f l ) ' ^"-"^(2 for the unknowns o>, o> r , or at least to determining whether or not these equations have a system of solutions such that co'/o is not xeal whenever g| 27 g\ is not zero If there exists a single system of solu- tions, there exist an infinite number of systems, but there appears to be no way of approach for a direct study of the preceding equations We can ainve at the solution of this problem in an indirect way by studying the inversion of the elliptic integral of the first kind Note. Let , <*' be two complex numbers such that U'/QJ is not real. The corre- sponding function p (u | , /) satisfies the differential equation 174 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 77 where g 2 and g s are defined by the equations (61) Por u = w, p (w) is equal to one of the roots e x of the equation 4p 8 2 p g z = When w varies from to w, p(it) describes a cuive L going fiom infinity to the point e l From the relation du = dp/V4 p 8 # 2 p ^ 3 we conclude that the half -period w is equal to the definite integral taken along the curve L An analogous expression for w' can be obtained by replacing e l by e 3 in the preceding integial We have thus the two half -periods expressed m terms of the invanants g^ <7 8 In order to be able to deduce from this result the solution of the problem before us, it would be necessary to show that the new system is equivalent to the system (61), that is, that it defines g% and g z as single-valued functions of w, w'. 78. The inverse function to the elliptic integral of the first kind. Let R(z) be a polynomial of the third or of the fourth degiee which is prime to its derivative. We shall write this polynomial in the form where a v a# a^ # 4 denote four different roots if R(z) is of the fourth degree. On the other hand, if R (z) is of the third degree, we shall denote its three roots by a is a a , & 8 , and we shall also set <z 4 =oo, agreeing to replace oo by unity in the expression R (&) The elliptic integral of the first kind is of the form where the lower limit # is supposed, for definiteness, to be different from any of the roots of R (#) and to be finite, and where the radical has an assigned initial value. If R () is of the fourth degree, the radical VjR(a) has four critical points a v a z , 8 , a 4 , and each of the determinations of ^/R(z) has the point & = oo for a pole of the second order. If R (z) is of the third degree, the radical VjR(js) has only three critical points in the finite plane a l} a, z a 8 , but if the variable z describes a circle containing the three points a l} a z , a s , the two values of the radical are permuted The point z = oo is therefore a branch point for the function V-R (s). Let us recall the properties of the elliptic integral u proved in 55. If u(z) denotes one of the values of that integral when we go from the point to the point & by a determined path, the same integral can take on at the same point # an infinite number of deter- minations which are included in the expressions (63) u = u(z)+2mv> + 2mV, u / u(z) + 2m<*> Ill, 78] INVERSE FUNCTIONS 175 if the path is varied In these foimulse m and w'are two entirely arbitraiy integers, 2 o> and 2o>' two periods whose ratio is not leal, and 7 a constant which we may take equal, for example, to the integral over the loop described about the point ^ Let p (u | <D, ft)') be the elliptic function constructed with the periods 2 a), 2 a/ of the elliptic integral (62). Let us substitute in that func- tion for the vanable u the integral (62) itself diminished by 7/2, and let (z) be the function thus obtained dz I >f l / M _ w >J *>( u 2"' (64) ^ v ~, ^ , , /^-rr % This function < (z) is a single-valued function of z In fact, if we leplace u by any one of the determinations (63), we find always, whatever m and m' may be, or which shows that <& (3) is single-valued Let us see what points can be singular points for this function & (z) First let x be any finite value of * different from a branch point. Let us suppose that we go fiom the point to the point z l by a definite path We arrive at z 1 with a ceitain value for the radical and a value u t for the integral In the neighborhood of the point z v 1/VJR() is an analytic function of z, and we have a development of the form Whence we derive (65) : If w t 7/2 is not equal to a period, the function p (u 7/2) is analytic in the neighborhood of the point u l9 and consequently <f (s) is analytic in the neighborhood of the point z r If ?/ t 7/2 is a period, the point u^ is a pole of the second order for p(u 7/2), and therefore * x is a pole of the second order for *(*), for in the neigh- borhood of the pomt u^ where P is an analytic function. 176 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 78 Suppose next that 2 approaches a cutical point a t In the neigh- borhood of the point a z we have where P t is analytic for z = a l} or whence, integrating term by teim, we find (66) u = it. If K, 1/2 is not a penod, p(u 1/2) is an analytic function of w in the neighboihood of the point w t Substituting in the develop- ment of this function in poweis of u v t the value of the difference u _ Vj obtained fiom the foimula (66), the fractional powers of (z _ # t ) must disappear, since we know that the left-hand side is a single-valued function of , hence the function $() is analytic in the neighboihood of the point a l Let us notice in passing that this shows that w t 1/2 must be a half-period Similarly, if w, 1/2 is equal to a period, the point a t is a pole of the first order for $(#) Finally, let us study the function $(z) for infinite values of z We have to distinguish two cases accoidmg as R(s) is of the fourth degiee or of the thud degree. If the polynomial R (z) is of the fouith degree, exterior to a circle C described about the origin as center and containing the four roots, each of the determinations of 1/Vj? (2) is an analytic function of l/ For example, we have for one of them and it would suffice to change all the signs to obtain the develop- ment of the second determination. If the absolute value of becomes infinite, the radical 1/VjJ (*) having the value which we have ]ust written, the integral approaches a finite value ^ w , and we have in the neighborhood of the point at infinity - 5-ft-ft- - If u^ 1/2 is not a period, the function p(u 1/2) is regular for the point u^ and consequently the point 2 = oo is an ordinary point for $ (z). If u n 1/2 is a period, the point . is a pole of the second m, 78] INVERSE FUNCTIONS 177 order for p (u 1/2), and since we can write, in the neighborhood of the point z = oo , the point & = oo is also a pole of the second ordei for the function <b(z) If R(z) is of the third degree, we have a development of the form which holds exterior to a circle having the ongin for center and containing the three critical points a v 2 , 8 It follows that (68) u = u m Reasoning as above, we see that the point at infinity is an ordi- nary point or a pole of the fiist oider for &(z). The function $(s) has certainly only poles for singulai points , it is therefore a rational function ofz } and the elliptic integral of the fiist kind (62) satisfies a relation of the form (69) where <E> (z) is a rational function We do not know as yet the degree of this function, but we shall show that it is equal to unity. For that purpose we shall study the inveise function In other words, we shall now consider u as the independent variable, and we shall examine the properties of the upper limit z of the integral (62), con- sidered as a function of that integral u We shall divide the study, which requires considerable care, into several parts 1) To every finite value of u correspond m values ofz if m is the degree of the rational function & (z) Tor let u^ be a finite value of u The equation $ (z) = p (u t 1/2) determines m values for z, which are in general distinct and finite, though it is possible for some of the roots to coincide or become infinite for particular values of u^ Let z l be one of these values of z The values of the elliptic integral u which correspond to this value of z satisfy the equation we have, then, one of the two relations u = u + 2m + 2m<', u = I- 178 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 78 In eithei ease we can make the vanable describe a path from # to gj such that the value of the integial taken ovei this path shall be precisely 2/ r If the function &(z) is of degree m, there aie then m values of z for which the integial (62) takes a given value u. 2) Let 11 ^ be a finite value of u to which corresponds a finite value ^ of z , that determination, ofz which approaches z 1 when u approaches ii l is an analytic function of u in the neighborhood of the point u^ For if z l is not a cutical point, the values of u and z which ap- proach respectively u t and z l are connected by the relation (65), where the coefficient a Q is not zero By the general theorem on implicit functions (I, 193, 2d ed , 187, 1st ed ) we deduce from it a development for & z 1 in positive integral powers of u u r If, for the particular value u^ z were equal to the critical value # t , we could in the same way consider the right-hand side of (66) as a development in poweis of Vs a t Since a is not zero, we can solve (66) for V# a,, and therefore for z a l9 expressing each of them as a power series in u 7/,. 3) Let u^ be one of the values which the integral u takes on when 1 2 1 becomes infinite , the point w w is a pole for that determination ofz whose absolute value becomes infinite In fact, the value of the integral u which approaches u^ is repre- sented in the neighborhood of the point at infinity by one of the developments (67) and (68) In the fiist case we obtain for 1/z a development in a series of positive powers of u u^ i = j S l (*- < )+ j 8 i ( tt _iO*+ , A^O; in the second case we have a similar development for 1/VS, and therefore . The point w is therefore a pole of the first or second order for z, according as the polynomial R(z) is of the fourth or of the third degree 4) We are going to show finally that to a value ofu there can cor- respond only one value of z. Tor let us suppose that as the variable z describes two paths going from # to two different points z v z s , the two values of the integral taken over these two paths are equal It would then be possible to find a path L joining these two points z v # 2 such that the integral HI, 78] INVERSE FUNCTIONS 179 would be zeio If we represent the integral n = X + Yi by tlie point with, the cooidmates (A", Y) m the system of rectangulai axes OX 9 OF, we see that the point u would describe a closed cuive F when the point s descubes the open cmve L. We shall show that this is not consistent with the propeities which we have ]ust demonstrated. To each value of u theie coi respond, by means of the relation p (11 1/2) = $ (z), a finite number of values of , each of which vanes in a continuous mannei with it, provided the path described by u does not pass through any of the points corresponding to the value & = oo * Accoiding to our supposition, when the variable u describes in its plane the closed curve T starting fiom the point A (U Q ) and returning to that point, describes an open aic of a con- tinuous cuive passing from the point ^ to the point # 2 Let us take two points Jl/and P (Pig. 28) on the curve r. Let the initial value of & at A be % v and let #', a" be the values obtained when we reach the points AT and P' respectively, after u has described the paths AM and A MNP. Again, let %" be the value with which we airive at the point P after u has described the arc AQP It lesults from the hypothesis that n and 2" are different Let us join the two points M and P by a transversal MP interior to the curve P 7 and let us suppose that the variable u describes the aic .eb/iJf and then the transversal MP , let i f be the value with which we arrive at the point P This value " W1 ^ be different fiom &" or else from #{'. If it is different from z[', the two paths AmMP and AQP do not lead to the same value of & at the point P If s' f and aj are different, the two paths AmMP and AmMNP do not lead to the same value at P ; therefore, if we start from the point M with the value z? for , we obtain different values for s according as we proceed from M to P along the path MP or along the path MNP In either case we see that we can replace the closed boundary r by a smaller closed bound- ary P t , partly interior to r, such that, when u describes this closed boundary, & describes an open arc. Repeating this same operation on the boundary T l9 and continuing thus indefinitely, we should obtain an unlimited sequence of closed boundaries r, T v T 2 , * having the same property as the closed boundary r, Since we evidently can * We assume the properties of implicit functions wbich will be established later (Chapter V) 180 SINGLE-VALUED ANALYTIC FUNCTIONS [m, 78 make the dimensions of these successive boundaiies appioach zeio, we may conclude that the boimdaiy T n appioaches a limit point X Prom the way in which this point lias been defined, there will always exist in the mtenoi of a ciicle of ladms e. described about X as a center a closed path not leading the vaiiable z back to its ongmal value, however small e may be Now that is impossible, for the point X is an oidinaiy point or a pole for each of the different determina- tions of 2 , in both cases z is a single-valued function of it, in the neighborhood of X We aie thus led to a contradiction in supposing that the mtegial fdz/'Vfi (2), taken over an open path L, can be zeio, or, what amounts to the same thing, by supposing that to a value of it coi res pond two values of z. We have noticed above that, if for two different values of # we have $ (z^) = & (#), we can find a path L from ^ to # 2 such that the integral - will be zero Hence the rational function <E> (z) cannot take on the same value foi two different values of z , that is, the function (z) must be of the first degree (z) = (az + b}/(cz -f d) It follows, fiom the relation (69), that I (70) * = - and we may state the following impoitant proposition : The upper limit & of an elliptic integral of the first kind, considered as a function of that integral, is an elliptic function of the second order Elliptic integrals had been studied in a thorough manner by Legendie, but it was by reversing the problem that Abel and Jacobi were led to the discovery of elliptic functions The actual determination of the elliptic function z=zf(ii) con- stitutes the problem of inversion By the relation (62) we have d* and therefore V72(s)=/ f (^). It is clear that the radical VTZfe) is itself an elliptic function of u We can restate all the preceding results in geometric language as follows Let R (z) be a polynomial of the third or fourth degree, prime to its derivative ; the coordinates of any point of the curve C 9 Ill, 78] INVERSE FUNCTIONS 181 (71) if = K(x), can be expressed in terms of elliptic functions of the integral of the first kind. J ; r x dx r* u= / = / J Xo y J Xo in such a way that to a point (x, y) of that curve corresponds only one value of u, any period being disregarded. To prove the last pait of the pioposition, we need only remark that all the values of u which coriespond to a given value of x aie included in the two expressions + 2 MjO) + 2 m 2 a/, I -f- 2 m l <*> + 2 All the values of u included in the first expression come from an even nurnbei of loops descubed about critical points, followed by the direct path, fiom X Q to x, with the same initial value of the radical V/2(ie) The values of u included in the second expression come from an odd number of loops described about the critical points, followed by the diiect path fiom X Q to x, wheie the corresponding initial value of the ladical ^/R (x) is the negative of the foimer If we aie given both x and y at the same time, the corresponding values are then included in a single one of the two formulae Fiom the investigation above, it follows that the elliptic function x =/(w) has a pole of the second order in a parallelogram if R(x) is of the third degiee, and two simple poles if E (x) is of the fourth degree , hence y =/ / (*0 1S ^ the third or of the fourth order, accord- ing to the degree of the polynomial R (ar) Note Suppose that, by any means whatever, the coordinates (x, y) of a point of the curve y*=tR(x) have been expressed as elliptic functions of a parameter v, say x = < (), y = ^(v). The integral of the first kind u becomes, then, The elliptic function f (tO/^i(*0 cannot have a pole, since u must always have a finite value for every finite value of v ; it reduces, then, to a constant A, and we have u = hv 4- 2 The constant I evidently depends on the value chosen for the lower limit of the integral u The coefficient k can be determined by giving to v a particular value. 182 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 79 79 A new definition of p(u) by means of the invariants. It is now quite easy to answer the question proposed in T7 Given two num- bers g 2> g s such that g$ 27^1 is not zero, there always exists an elliptic function p(w) for which g^ and ^ are the invariants Tor the polynomial -R(s)=4**-0V* $s is prime to its derivative, and the elliptic integral /<fe/V.R(s) has two periods, 2 o>, 2 ', whose ratio is imaginary. Let p (u , <J) be the eoi responding elliptic function We shall substitute for the aigu- rnent u in this function the integial (72) -JT, where H is a constant chosen in such a way that one of the values of u shall be equal to zero for s = o> We shall take II, for example, equal to the value of the integial f^d/^/R (c) taken over a ray L starting at # We shall show fiist that the function thus obtained is a single- valued analytic function of z. Let & be any point of the plane, and let us denote by v and v* the values of the integrals starting with the same initial value for -*jR(z) and taken over the two paths zjriz, zjiz, which together form a closed curve containing the three critical points of the radical Consider the closed curve FIG. 29 formed by the curve z^mznz^ the segment ^, the circle C of very large radius, and the segment Zz^ The function 1/V.ft (z) is analytic in the interior of this boundary, and we have the relation , i JL. tf _ which becomes, as the radius of the circle C becomes infinite, Hi, 79] INVERSE FUNCTIONS 183 The values of u resulting from the two paths z Q ?nz, z Q nz theiefore satisfy the relation u + u' = 0. From this we conclude that the function is a single-valued function of z We have seen that it is a linear function of the form (az -{- ft) /(ess -{- d) To determine a, 1>, c, d it will suffice to study the development of this function in the neigh- "boihood of the point at infinity We have in this neighborhood /*() 2z^~ 4 - 2 4* 1 / 2*J 16*1 ' hence the value of u> which is zero for z infinite, is represented by the series whence It follows that the difference p(ju) s is zeio f or = oo But the difference (az -f- V)/(cz> + d) z can be zero for z = oo only if we have c = ? Z> = 0, a = d, and the function p(w| o>, <o r ) reduces to when we substitute for u the integral (72) Taking the point at infinity itself for the lower limit, this integral can also be written in the form (72() M ' and this relation makes p (u) = #, where the function p (&) is con- structed with the periods 2 o> 3 2 o> r of the integral fdz/"VR (z). Comparing the values of du/dz deduced from these relations, we have p'(w) = VjR(), or, after squaring both sides, (73) p(*) = *(*) The numbers ^ 2 , g^ therefore, are the invanants of the elliptic func- tion p(u), constructed with the periods 2 o>, 2 <o'. This result answers the question proposed above in 77 If g% 27 g\ is not zero, the equations (61) are satisfied by an infinite number of systems of values for <o, o) r . If e v e# 8 are the three roots of the equation 184 SINGLE-TALUED ANALYTIC FUNCTIONS [III, 79 one system of solutions is given, for example, by the formulae , (74) 0, v ; from which all other systems will be deducible, as has been explained In the applications of analysis m which elliptic functions occui, the function p (u) is u&ually defined hy its invariants In ordei to carry through the numerical computations, it ib necessary to calculate a pair of periods, knowing g 2 and gr 3 , and also to be able to find a root of the equation p (u) = A, where A is a given constant Poi the details of the methods to be followed, and for information regaidmg the use of tables, we can only lefei the leadei to special treatises * 80. Application to cubics in a plane. When pf 27 g\ is not zero, the equation (75) f = x*-g^-~g z repiesents a cubic without double points This equation is satisfied by putting x = p(w), y = p'(w)> wheie the invariants of the function p(u) aie piecisely g 2 and g^ To each point of the cubic coi responds a single value of u in a suitable paiallelogiam of periods For the equa- tion p (M) = x has two roots x and u z in a parallelogram of periods, the sum u^ + it 2 is a period, and the two values p'(u ) and p'(^ 2 ) are the negatives of each other They are therefore equal lespectively to the two values of y which correspond to the same value of x In general, the coordinates of a point of a plane cubic without double points can be expressed by elliptic functions of a parameter. We know, in fact, that the equation of a cubic can be reduced to the form (75) by means of a projective transfoirnation, but this transformation cannot be effected unless we know a point of inflec- tion of the cubic, and the determination of the points of inflections depend upon the solution of a ninth-degree equation of a special form We shall now show that the parametric representation of a cubic by means of elliptic functions of a parameter can be obtained without having to solve any equation, provided that we know the cooidmates of a point of the cubic Suppose first that the equation of the cubic is of the form (76) f = a o o* + 86^ + 3 b 2 x + ft,, * The formulae (39) which give the development of <r (u) in a power series, and those which result from it by differentiation, enable us, at least theoretically, to calculate <r (u) , <r'(w) , <r"(u) , and consequently (u) and p (u) , for all systems of values Ill, 80] INVERSE FUNCTIONS 185 m which case the point at infinity is a point of inflection This equation can be reduced to the preceding foiin by putting y = - x = &!/&() + 4xy& , which gives where the invariants g 2 , g z are given by the formulse Hence we obtain for the coordinates of a point of the cubic (76) the following formulse Let us now consider a cubic C^ and let (or, /?) be the coordinates of a point of that cubic The tangent to the cubic at this point (#, ft) meets the cubic at a second point (', )8 f ) whose coordinates can be obtained rationally If the point (or', ') is taken as origin of coor- dinates, the equation of the cubic is of the form ttfa y) + h where < t (x 5 y) denotes a homogeneous polynomial of the i th degree (L = 1, 2, 3) Let us cut the cubic by the secant y = for , then a; is determined by an equation of the second degiee, ^C 1 . 0+ *i(li 3= whence we obtain where R(t) denotes the polynomial <#j(lj 4 < 8 (1, t} ^(1, ), which is in general of the fourth degree The roots of this polynomial are precisely the slopes of the tangents to the cubic which pass through the oiigm * We know a priori one root of this polynomial, the slope of the straight line which joins the origin to the point (a, fi). Putting t = + 1/t', we find where the polynomial Rft) is now only of the third degree. The coordinates (#, y) of a point of the cubic C 8 are therefore expressible rationally in terms of a parameter t 1 and of the square root of a *Two roots cannot be equal (see Vol I, 103, 2d ed , 108, 1st ed ) -TRANS, 186 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 80 polynomial R,(t') of the third degree We have ]ust seen how to express t' and Vjf? 1 (^') as elliptic functions of a parameter u , hence we can expiess x and y also as elliptic functions of u It follows fioin the nature of the methods used above that to a point (r, y) of the cubic coirespond a single value of t and a definite value of -vR(), and hence completely detei mined values of t 1 and V-R^tf 1 ) Now to each system of values of t 1 and VTt^') corre- sponds only one value of w in a suitable parallelogiam of peiiods, as we have already pointed out The expiessions x =f(ii), y =f 1 (u), obtained for the cooidinates of a point of C 3 , are therefore such that all the determinations of u which give the same point of the cubic can be obtained from any one of them by adding to it various periods. This parametric repiesentation of plane cubics by means of elliptic functions is very important * As an example we shall show how it enables us to deter- mine the points of inflection Let the expressions for the coordinates be x =/(u), y =,/i(tt) , the arguments of the points of intersections of the cubic with the straight line Ax -f By + C = are the roots of the equation Since to a point (x, y) corresponds only one value of u in a parallelogram of periods, it follows that the elliptic function Af(u) + Bf^u) + C must be, in general, of the third order The poles of that function are evidently independent of -d, B, C , hence if u v u z , u s are the three arguments corresponding respec- tively to the three points of intersections of the cubic and the straight line, we must have, by 68, u where X is the sum of the poles in a parallelogram Replacing u by JBT/3 + u in/(w) and /^w), the relation can be written m the simpler form % + M 2 + u 3 = period Conversely, this condition Is sufficient to insure that the three points M l (u=w 1 ), If 2 (u = w 2 ), Jf 8 (u = u 3 ) on the cubic shall lie on a straight line For let M'% be the third point of intersection of the straight line Jf x J9f s with the cubic, and u$ the corresponding argument Since the sum u 1 +u 2 + u% is equal to a period, tCj and u differ only by a period, and consequently M'% coincides with M 8 If u is the value of the parameter at a point of inflection, the tangent at that point meets the curve in three coincident points, and 3w must be equal to a period We must have, then, u = (2m 1 w + ^m^^/B All the points of inflec- tion can be obtained by giving to the integers m l and m a the values 0, 1, 2. Hence there are mne points of inflections. The straight line which passes through *CLEBSCH, Ueber dtejent&en Curven, deren Coordinatensich als elhptwche JFVnc- twnen e^nes Parameters darstelten lassen (Crelfe's Journal, Yol Ill, 81] INVERSE FUNCTIONS 187 the two points of inflection (Zm^w + 2m z b>')/3 and (2mj[w 4- 2m2w')/3 meets the cubic in a third point whose argument, ^u -f- 2(m 2 + mgX 3 is again one thud of a period, that is, in a new point of inflection The number of straight lines which meet the cubic in three points of inflection is theiefore equal to (9 8)/(3 2), that is, to twelve Note The points of mteisection of the standard cubic (75) with the straight line y = mz -f n are given by the equation p'(u) mp (u) n = 0, the left-hand side of which has a pole of the thud older at the point u = The sum of the arguments of the points of intersection is then equal to a period If u and u z are the aiguments of two of these points, we can take w t u z for the argu- ment of the third point of intersection, and the abscissas of these three points are respectively p (uj, p (u 2 ), p (w t + U Q ) We can deduce from this a new proof of the addition formula for p (u) In tact, the abscissas of the points of inter- section are roots of the equation 4x 3 - gr 2 oj - 3 = (ma? + n) z , hence m 2 x t + OJ 2 + JC 3 = p(u l ) + p(ua) + pK + u a ) = ~ On the other hand, from the straight line passing through the two points Jf^t^), M 2 (w 2 ), we have the two relations p'(u^=mp (u : ) + n, p'(u%) =mp (u 2 ) + w, whence and this leads to the relation already found in 74, 81. General formulae for parameter representation. Let R (x) be a polynomial of the fourth degree pi line to its derivative. Considei the curve C 4 repiesented by the equation (77) f = ^(a;)= a x* + 4a t a; 8 + 6^8? + 4a t as -f a^ We shall show how the coordinates x and y of a point of this curve can be expressed as elliptic functions of a parameter. If we know a root a of the equation R (x) = 0, we have already seen in the treat- ment of cubics how to proceed. Putting x = a + 1/x' 9 the relation (77) becomes where Rfa') is a polynomial of the third degree. Hence the curve (7 4 , by means of the relations x = a -j- I/a; 1 , y = 2/ /' 2 , corresponds point for 188 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 81 point to the curve Cg of the third degree whose equation is y^Rfa'} Now x' and y } can be expiessed by means of a parameter u, in the foim x 1 = <rp(w)4-/3, if = ccp l (ii), by a suitable choice of a, /3 and of the invariants of p(n) We deduce fiom these relations the following expressions for x and y : whence we find <7tf = dr/y, so that the parameter u is identi- cal, except for sign, with the integral of the fiist kind, fdx/^/R (x), and the formulae (78) constitute a generalization of the results for the simple case of parametric representation in 80 Let us considei now the general case in which we do not know any loot of the equation J2(;r) = We are going to show that x and y can be expressed rationally in terms of an elliptic function p(u) with known invariants, and of its derivative p'(&), without introducing any other irrationality than a square root. Let us replace for the moment x and y by t and v icspectively, so that the relation (77) becomes (77') v* = R (t) = a^ + 4 a t f + 6 aj? + 4 a z t + a^ The polynomial R (t) can be expressed in the form in an infinite numbei of ways, where < 1? < 2 , < 3 aie polynomials of the degrees indicated by then subscripts. For let (a, /$) be the cooi- dinates of any point on the curve C 4 Let us take a polynomial <f> 2 (t) such that 4> 2 (a) = /3, which can be done in an infinite number of ways ; then the eqnafaon will have the root t = a, and we can put ^(tf) = t a The poly- nomial R (t) having been put in the preceding f oim, let us consider the auxiliary cubic C 8 represented by the equation (79) If we cut this cubic by the secant y = tx, the abscissas of the two variable points of intersection are roots of the equation and can be expressed in the form Ill, 81] INVERSE FUNCTIONS 189 where v is determined by the equation (77') Conversely, we see that t and v can be expiessed lationally in terms of the coordinates a, y of a point of C 8 by the equations (80) ,.J. .- Now cc and y can be expressed as elliptic functions of a parameter , since we know a point on the cubic <? 3 that is the ongin Then t and v can also be expiessed as elliptic functions of u The method is evidently susceptible of a great many vai rations, and we have mtio- duced only the irrational ft = V/2 (a), wheie a is aibitiaiy. We are going to carry thiough the actual calculation, supposing, as is always admissible, that we have first made the coefficient a t of t* disappeai in R(). We can then write a Q R (t) = (a/) 2 + 6 V / + 4 a Q a & t + a a 4 and put ^(0 = -!, < 2 (0=V 2 > ^(O^K^ + 'KV + VV The auxiliary cubic C g has the form (81) 6 a a 2 itf + 4 a^aPy + a^af + 2 a Q f - r = 0. Following the general method, let us cut this cubic with the secant y~tx, the equation obtained can be wntten in the form - 2 a/ - (6 vr/ + 4 a a,# + a A ) = 5 \J!J/ <lt whence we obtain i = a *+VV20). Conversely, we can express # and V 22 (#) in terms of a; and ^: (82) * = J, V^(0 = ^- On the other hand, solving the equation (81) for y, we have - 2 g Q a s r 2 + V4 gajx 4 - x(a a 4 x 2 -1) (6 g a> g g + 2 Q The polynomial under the radical has the root a; = 0. Applying the method explained above, we can then express x and y as elliptic functions of a parameter. Doing so, we obtain the results 190 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 81 where the mvaiiants #,, <7 8 of the elliptic function p(u) have the following values (84) ?*=-a> ' ff ' = Substituting the preceding values for x and y in the expressions (82), we find (85) '-2 =vz *P()-?-7 We can write these results in a somewhat simpler form by noting that the relations (86) are compatible accoiding to the values (84) of the invariants g^ and g^ On the other hand, we can substitute for its equivalent p (u +_v)+ p (u) + p (v). Combining these results and replacing # and Vj(tf) bj a? and y respectively, we may formulate the result in the following proposition . The coordinates (x, y) of any point on the curve C 4 , represented by the equation (77) (ivhere a^ = 0), can be expressed in terms of a varir able parameter u by the formuUe (87) where the invariants g^ and g^have the values given by the relations (84), and where p(t?), p'(v) are determined by the compatible equations (86) !From the formula (45), established above ( 74), we derive, by differentiating the two sides of that equality, 1 d Ill, 82] INVERSE FUNCTIONS 191 that is, dx/du = y/^fa^ or du = [Vaj/y] f7x. The parameter it, there- fore, represents the elliptic integral of the fiist kind, Vo^JWa/ V.R (x), and the formulae (87) furnish the solution of the generalized prob- lem of parameter representation. 82. Curves of deficiency one. An algebraic plane curve C n of degree n cannot have more than (n 1) (n 2)/2 double points without degenerating into seveial distinct curves. If the curve C n is not degenerate and has d double points, the difference ***~ <T d is called the deficiency of that curve. Curves of deficiency zero are called unicursal curves , the cooidmates of a point of such a carve can be expressed as rational functions of a parameter The next simplest curves are those of deficiency one; a cuive of deficiency one has (n !)(% 2)/2 1 = n(n 3)/2 double points. The coordinates of a point of a curve of deficiency one can be expressed as elliptic functions of a parameter. In order to prove this theorem, let us consider the adjoint curves of the (n 2)th order, that is, the curves C n _ 2 which pass through the n(n 3)/2 double points of C n . Since (n 2) (n + 1)/2 points are necessary to determine a curve of the (n 2)th degree, the adjoint curves C R _ 2 depend still upon arbitrary parameters If we also require that these curves pass through n 3 other simple points taken at pleasure on C n) we obtain a system of adjoint cuives which have, in common with C n , the n(n 3)/2 double points of C n and n 3 of its simple points Let F(x 9 y)= be the equation of C n , and let be the equation of the system of curves C r w _ 2J where X and p are arbi- trary parameters. Any curve of this system meets C n m only three variable points, for each double point counts as two simple points, and we have n ( n - 3) 4- n 3 = n (n - 2) - 3. Let us now put 192 SINGLE-VALUED ANALYTIC FUNCTIONS [III, 82 when the point (a, y) desciibes the curve C n , the point (x\ ?/') de- scubes an algebraic cuive C' whose equation would be obtained by the elimination of a: and y between the equations (88) and F(x, y) = The two cuives C 1 and C n eouesponcl to each other point for point by means of a birational transformation This means that, con- versely, the cooidinates (it, y) of a point of C n can be expiessed lationally in teims of the cooidinates (&', y 1 ) of the corresponding point of C" To prove this we need only show that to a point (# f , ?/') of C 1 there coriesponds only one point of C n , or that the equations (88), togethei with F(x, y) = 0, have only a single system of solu- tions foi a and #, which vaiy with x 1 and y\ Suppose that to a point of C' there coriespond actually two points (a, &), (#', ') of C n which are not among the points taken as the basis of the system of curves C n ^ 2 . Then we should have a, J) fja, 5) and all the curves of the system which pass through the point (a, V) would also pass thiough the point (a\ &') The curves of the system which pass thiough these two points would still depend linearly upon a variable parameter and would meet the curve C n in a single variable point The coordinates of this last point of intei section with C n would then be lational functions of a variable parameter, and the curve C n would be umcursal. But this is impossible, since it has only n(n 3)/2 double points Hence to a point (x', y') of C' corresponds only one point of C n? and the coordinates of this point are, by the theory of elimination, rational functions of x r and y 1 (89) x = <>', ?/'), y = <i> 2 (x', y') In order to obtain the degree of the curve C 1 , let us try to find the number of points common to this curve and any straight line ax 1 + ly* + o = This amounts to finding the number of points common to the curve C n and the curve since to a point of C' corresponds a single point of C n , and conversely. Now there are only three points of intersection which vary with a, b, c. The curve C 1 is therefore of the third degree To sum up, the coor- dinates of a point of the curve C n can be expressed rationally in terms of the coordinates of a point of a plane cubic , and since the coordinates of a point of a cubic are elliptic functions of a parameter, the same thing must be true of the coordinates of a point of C n HI, Exs ] EXERCISES 193 It results also from the demonstration, and from what has been seen above for cubics, that the representation can be made in such a way that to a point (* 7 y) of C n corresponds only one value of it in a paiallelogram of periods Let x = ^(M), y = ^(w) be the expressions for x and y denved above , then every Abehaii integial w = fH (#, ?/) dx associated with the curve C n (I, 103, 2d ed , 108, 1st ed.) is reduced by this change of vanables to the integral of an elliptic function , hence this integial w can be expressed m teims of the transcendental^ p, a- of the theory of elliptic functions The mtioduetion of these tian- scendentals in analysis has doubled the scope of the integral calculus Example Bicircular quwtics A cuive of the fourth degree \\ ith two double points is of deficiency one If the double points aie the circulai points at in- finity, the curve C 4 is called a bicircular quartic If we take foi the ongm a point of the curve, we can take foi the adjoint cuives O n -z cncles passing thioughtkeorigm In order to have a cubic corresponding point for point to the quartic C 4 , vi e need only follow the general method and put x' /(z 2 H- y 2 ), y' = y/(x 2 + y 2 ) We have, conversely, x = ir'/^' 2 -f 2/' 2 ), y y'Jtf* + y' 2 ) These formulae define an inversion with respect to a cucle of unit radius described with the oiigm as centei To obtain the equation of the cubic Cg, it \vill sumee to replace x and y in the equation of C 4 by the preceding values Suppose, foi example, that the equation of the quartic C 4 is (x 2 + y 2 ) 2 ay = 0, the cubic Cj will have for its equation ay'ty* + ' 2 ) 1=0 Note When a plane curve C n has singular points of a higher order, it is of deficiency one, provided that all its singular points are equivalent to n(n 3)/2 ordinary double points For example, a curve of the fourth degree having a single double point at which two branches of the cuive are tangent to each other without having any other singularity is of deficiency one ; to verify this it suffices to cut the quartic by a system of comes tangent to the two branches of the quartic at the double point and passing through another point of the quartic. The curve y^ = 12 (jc), where E(x) is a polynomial of the fourth degree pume to its denvative, has a smgulanty of this kind at the point at infinity. It is reduced to a cubic by the following birational transformation x = a;', y = from which at is easy to obtain the formulae (87) EXERCISES 1 Prove that an integral doubly periodic function is a constant by means of the development (The condition f(z + &0 =/() requires that we have A n = If n & ) 194 SINGLE-VALUED ANALYTIC FUNCTIONS [m,Exs 2 If fit is not a multiple of TT, we have the formula a mr/ (Change z to z + a in the expansion for ctn z, then integrate between the limits and z ) 3 Deduce from the preceding result the new infinite products a) / a \ cos a \ Sa-ffly-t-JL 2a (2n ] smo: sing sin a cos z cos a 1 cos a Transform these new products into products of pnmaiy functions or into products that no longer contain exponential factors, such as -**\. Pi 97rV L T*/\ 97TV f 4. Derive the relations tanz = 2z| : h- Establish analogous relations for 1 1 , .^ sin z sin a cos z cos a 5. Establish the relation ^i-g!. 6 Decompose the functions 1 P"(M) ' into simple elements. 7. If flT 2 = 0, we have p (CM ; 0, 8 ) = op (u ; 0, gr 8 ), tf( m , 0, f/ 8 ) = p'( w , 0, ^ 8 ), where a is one of the cube roots of unity. From this deduce the decomposition 'K)- p'()] into simple elements when g z = Ill, Exs ] EXERCISES 195 8 Given the integrals /ax + b f ax 2 + 6 -=dx+ I , (%c, (x-l)Vx 3 -! ^ J Vl + x 4 ' /dx r ax 2 + 6 x 3Vx3-x* ^ V(1 JC 3 )(1 ^x 2 )^' it is required to express the variable x and each one of these integrals in terms of the transcendentals p, f, a- 9 Establish Heimite's decomposition formula ( 73) by equating to zero the sum of the residues of the function F(z)[(x, z) (Z Q z)l m a paral- lelogiain of penods, where F(x) is an elliptic function and where a, x are considered as constants 10 Deduce fiom the formula (00) the relation 77 = 0'"(0)/12 0'(0). (It should be noticed that the series f 01 <r (u) does not contain any terms in w 8 ) 11* Expiess the coordinates x and y of one of the following curves as elliptic functions of a parameter y=^L[(-a) (x-6) <s-c)p, y = ^[(as-a) (x-5)] 2 , y* = ^. (sc a) 2 (z 6) 8 (a; c) 3 , 1^ = ^ (x a) 2 (x 6) 3 , y*=ul(x-a)(-6), 2/ 6 =^L (x-a) 8 (x-6) 4 (-c) 5 , 3^ = ^. (x a) 8 (x - 6) 4 , 2/6=-d. (x a) 8 (aj &) 5 , y 6 = ^d (x a) 4 (a: 6) 5 , - a) (x - &) ( c)] = 0, The variable parameter is equal, except for a constant, to the integral /(1/y) <Zx. [BRIOT ET BOCQUET, Th&orie desfoncltons doublement pfriodzques, 2d ed , pp 388-412.] CHAPTER IV ANALYTIC EXTENSION I DEFINITION OF AN ANALYTIC FUNCTION BY MEANS OF ONE OF ITS ELEMENTS 83. Introduction to analytic extension. Let f(z) be an analytic func- tion in a connected poition A of the plane, bounded by one or more cuives, closed 01 not, wheie the word curve is to be undei stood in the usual elementaiy sense as heretofore If -we know the value of the function f(z) and the values of all its successive derivatives at a definite point a of the region A, we can deduce from them the value of the function at any othei point b of the same region To piove this, ]oin the points a and I by a path L lying entnely in the region A , for example, by a bioken line 01 by any foim of cuive whatever. Let 8 be the lower limit of the dis- tance from any point of the path L to any point of the boundary of the region A, so that a ciicle with the radius 8 and with its centei at any point of L will lie entirely in that region By hypothesis we know the value of the function f(a) and the values of its successive derivatives f r (a), /"(&), , for & = a We can therefore write the power series which represents the function /(#) in the neighborhood of the point a . (1) The radius of convergence of this series is at least equal to 8, but it may be greater than 8 If the point I is situated in the circle of convergence C of the preceding senes, it will suffice to replace by b in order to have/(Z>) Suppose that the point b lies outside the circle C , and let <Xi be the point where the path L leaves C * (Fig. 30) Let us take on this path a point 1 within C and near a v so that the * Since the value of f(z) at the point 6 does not depend on the path so long as it does not leave the region A, we may suppose that the path cuts the ciicle GQ in only one point, as in the figure, and the successive circles Cj, C" 2 , in at most two points This amounts to taking for ^ the last point of intersection of L and <? , and similarly for the others 196 IV, 83] ELEMENTS OF AN ANALYTIC FUNCTION 197 distance between the two points ^ and ^ shall be less than S/2 The series (1) and those obtained from it "by successive differentiations enable us to calculate the values of the function f(z) and of all its derivatives, /(*,), f(*J, . . , /<) ( gj ), . , f or * = * r The coefficients of the series which represents the function /(*) in the neighborhood of the point z l are therefore determined if we know the coefficients of the fiist series (1), and we have m the neighborhood of the point ^ The radius of the circle of convergence C^ of this series is at least equal to 8; this circle contains, then, the point <K : within it, and there is also a part of it out- side of the circle C . If the point I is in this new circle C v it will suffice to put z = # in the series (2) in order to have the value of /(). Sup- pose that the point I is again outside of C v and let a 2 be the point where the path z^b leaves the circle. Let us take on the path L a point # 2 within C l and such that the distance between the two points # 2 and a 2 shall be less than 5/2 The series (2) and those which we obtain from it by successive dif- ferentiations will enable us to calculate the values of f(z) and its derivatives /(*,), /(^ f'(*d> * * * at tiie P omt form a new series, (3) O + which represents the function f(&) in a new circle C 2 with a radius greater than or equal to 8. If the point b is in this circle C 2 , we shall replace # by b in the preceding equality (3); if not, we shall continue to apply the same process. At the end of a finite number of such operations we shall finally have a circle containing the point b within it (in the case of the figure, b is in the interior of C^) , for we can always choose the points v 3 , # 8 , - in such a way that the dis- tance between any two consecutive points shall be greater than S/2. On the other hand, let 5 be the length of the path . The length of 198 ANALYTIC EXTENSION [IV, 83 the broken line az^ 2 ^ p -^ p is always less than S 9 hence we have ^s/2 + \z p b\<S Let p be an integer such that (p/2 + 1) 8 > S The piecedmg inequality shows that aftei p opeiations, at most, we shall come upon a point z p of the path L whose distance from the point I will be less than 8, the point b will be in the mtenor of the circle of convergence C f of the power series which represents the function /() in the neighboihood of the point p) and it will suffice to replace # by I in this series in oider to have/(Z>) In the same way all the derivatives /'(), /"(&), * can be calculated The above reasoning proves that it is possible, at least theoretically, to calculate the value of a function analytic in a region A, and of all its derivatives at any point of that region, provided we know the sequence of values, of the function and of its successive derivatives at a given point a of that region It follows that any function analytic in a region A is completely determined in the whole of that region if it is known in a region, however small, surrounding any point a taken in A, or even if it is known at all points of an arc of a curve, however shoit, ending at the point a For if the function f(z) is determined at every point on the whole length of an arc of a curve, the same must be true of its derivative /'(), since the value f(zj at any point of that arc is equal to the limit of the quotient [/(V,) /(X)]/^ x ) when the point z 2 appi caches z 1 along the arc considered , the deriv- ative/'^) being known, we deduce from it in the same way /"(), and from that we deduce /"'(*), All the successive derivatives of the function /() will then be detei mined for # = a. We shall say for brevity that the knowledge of the numerical values of all the terms of the sequence (4) determines an element of the function /(). The result reached can now be stated in the following man- ner : A function analytic in a region A is completely determined if we know any one of its elements We can say further that two func- tions analytic in the same region cannot have a common element without being identical. We have supposed for definiteness that the function considered, /(), was analytic in the whole region; but the reasoning can be extended to any function analytic in the region except at certain singular points, provided the path L, followed by the variable in going from a to 5, does not pass through any singular point of the function. It suffices for this to break up the path into several arcs, IV, 84] ELEMENTS OF AN ANALYTIC FUNCTION 199 as we have already done ( 31), so that each one can be inclosed in a closed boundary inside of which the branch of the function /( ^) considered shall be analytic The knowledge of the initial element and of the path described by the variable suffices, at least theoieti- cally, to find the final element, that is, the numencal values of all the terms of the analogous sequence 84. New definition of analytic functions. Up to the present we have studied analytic functions which were defined by expressions which give their values for all values of the variable in the field in which they were studied. "We now know, fiom what precedes, that it is possible to define an analytic function for any value of the variable as soon as we know a single element of the function ; but in order to present the theory satisfactorily from this new point of view, we must add to the definition of analytic functions accoiding to Cauchy a new convention, which seems to be woith stating in considerable detail. Let/ x (), f 2 (z) be two functions analytic respectively in the two regions A I} A having one and only one part A* in common (Fig. 31) If in the com- mon part A r we have / 2 (s) = / 1 (s), which will be the case if these two functions have a single common element in this region, we shall regard f^z) and / 2 (^) as forming a single function F(z), analytic in the region A^AQ, by means of the following equalities: *(*) =/i(*) m A v and F (*)=/2(*) in A r We shall also say that f z (&) is the analytic extension into the region A 2 A' of the analytic f unction ffa), which is supposed to be defined only in the region A^ It is clear that the analytic extension of / t () into the region of A Z exterior to A l is possible in only one way.* *In order to show that the preceding convention is distinct from the definition of functions analytic m general, it suffices to notice that it leads at once to the following consequence IJ a f unction f(z) i& analytic in a region A, every other analytic func- tion f fa), under these conventions, which coincides withf(z) in a part of the region A is identical withf(z) in A Now let us consider a function F(z) defined for all values of the complex variable z in the following manner F(z) sin , if z 5* -> F\^J = However odd this sort of convention may appear, it has nothing in it contra- dictory to the previous definition of functions m general analytic. The function thus defined would he analytic for all values of z except for z = ?r/2, which would 200 ANALYTIC EXTENSION [IT, 84 Let us now consider an infinite sequence of numbers, real or imaginary, (6) ; a v a z > 3 w ? - 9 subject to the single condition that the series CO o + V + ^ +' + a *" + * ' ' converges for some value of 2 different from zero (We take 2 = for the initial value of the variable, which does not in any way restrict the generality.) The series (7) has, then, by hypothesis, a cucle of convergence C Q whose radius R is not zero If R is infinite, the series is convergent for every value of # and represents an inte- gral function of the variable. If the radius R has a finite value dif- ferent from zero, the sum of the series (7) is an analytic function f(z) in the interior of the circle C But since we know only the sequence of coefficients (6), we cannot say anything a priori regard- ing the natuie of the function outside of the cucle (7 We do not know whether or not it is possible to add to the circle C f an adjoin- ing region forming with the circle a connected region A such that there exists a function analytic in A and coinciding with/(^) in the interior of C , The method of the preceding paragiaph enables us to determine whether this is the case or not Let us take in the circle C a point a different from the origin By means of the series (7), and the senes obtained from it by term-hy-term differentiation we can calculate the element of the f unction /() which eoi responds to the point a, and consequently we can form the power series (8) which represents the function/^) in the neighborhood of the point a This series is certainly convergent m a circle about a as center with a radius R a\ ( 8), but it may be convergent in a larger circle whose radius cannot exceed R + \a\. Por if it were convergent in be a singular point of a particular nature But the properties of this function F(z) would be in contradiction to the convention which, we have just adopted, since the two functions F(z) and sm z would be identical for all the values of z except for z = ir/2, which would be a singular point for only one of the two functions Weierstrass, in Germany, and Meray, in France, developed the theory of analytic functions by starting only with the properties of power series, their investigations are also entirely independent Meray's theory is presented m his large treatise, Lemons nouveltes $ur F Analyse tttfimtesimale It is shown in the text how we can define an analytic function step by step, knowing one of its elements but always supposing known the theorems of Cauchy on analytic functions IV, 84] ELEMENTS OF AN ANALYTIC FUNCTION 201 a circle of radius R -f \a\ + S, the senes (7) would be con vei gent in a cucle of ladius R -f- 8 about the origin as center, contiary to the hypothesis. Let us suppose first that the radius of the circle of con- vergence of the series (8) is always equal to R ||, wheiever the point a may be taken in the circle C Then there exists no means of extending the function /fc) analytically outside of the ciicle, at least if we make use of power senes only We can say that there does not exist any function F(z) analytic in a region A of the plane gieater than and containing the cncle C Q and coinciding with f(z) in the circle C , for the method of analytic extension would enable us to determine the value of that function at a point exterior to the circle C , as we have ]ust seen The cucle C is then said to be a natural "boundary foi the function f(?) Furthei on we shall see some examples of this Suppose, in the second place, that with a suitably chosen point a in the cucle C the cucle of conveigence C^ of the series (8) has a radius greater than R \ a \ G This circle C l has a part exterior to C (Fig 32), and the sum of the senes (8) is an analytic function / x () in the circle C l In the interior of the circle y with the center a, which is tangent to the circle C internally, we have /!()=/(*) (8); hence this equality must subsist in the whole of the region common to the two circles C , C l The senes (8) gives us the analytic extension of the function /() into the portion of the circle C l exterioi to the cucle C Let a* be a new point taken in this region , by proceeding in the same way we shall form a new power senes in powers of & a', whi6h will be con- vergent m a circle C 2 If the cucle C 2 is not entirely within C 1? the new senes will give the extension of /() in a more extended region, and so on in the same way. We see, then, how it is possible to extend, step by step, the region of existence of the function /(), which at first was defined only in the interior of the ciicle C . It is clear that the preceding process can be carried out in an in- finite number of ways In order to keep in mind how the extension was obtained, we must define precisely the path followed by the 202 ANALYTIC EXTENSION [IV, 84 variable Let us suppose that we can obtain the analytic extension of the function denned by the series (7) along a path L, as we have just explained Each point x of the path L is the center of a circle of conveigence of radius r in the mteiior of which the function is lep- resented by a con vei gent senes arranged in powers of z x The radius r of this circle varies continuously with x For let x and x 1 be two neighboung points of the path L } and r and r' the corresponding radii If x r is near enough to x to satisfy the inequality \x' x \ < r, the radius r' will lie between r | x 1 x \ and r + \x' x |, as we have seen above Hence the difference r f r appi caches zero with \x* x \ Now let C' Q be a cucle with the ladius R/2 descubed with the origin as center; if a is any point on the circle C' QJ the ladius of conver- gence of the senes (8) is at least equal to 7?/2, but it may be greater Since this iadms vanes in a continuous manner with the position of the point a, it passes thiough a minimum value R/2 + r at a point of the circle C' Q We cannot have r > 0, for if r weie actually posi- tive, theie would exist a function F(z) analytic in the circle of radius 22 _j_ r about the ongin as center and coinciding with /(&) in the interior of C For a value of z whose absolute value lies between R and R + r, F(z) would be equal to the sum of any one of the series (8), where a is a point on CJ such that [ z a \ < R/2 + r Accoiding to Cauchy's theoiem, F(z) would be equal to the sum of a power series convergent in the circle of radius R + r, and this series would be identical with the senes (7), which is impossible. There is, therefore, on the circumference of C' at least one point a such that the circle of convergence of the series (8) has R/2 for its iadms, and this circle is tangent internally to the cncle C Q at a point a where the radius Oa meets that circle The point <z is a singular point of /(#) on the circle C In a cucle c with the point a for center, however small the radius may be taken, there cannot exist an analytic function which is identical with/(#) in the part common to the two circles C Q and c. It is also clear that the circle of conver- gence of the sferies (8) having any point of the radius Oa foi center is tangent internally to the circle C at the point a * * If all the coefficients a* of the series (7) are real and positive, the point z-Ris necessarily a singular point on C7 In fact, if it were not, the power senes which represents/ (2) in the neighborhood of the point z = R/2, would have a radius of convergence greater than jR/2 The same would he true a fortiori of the series IV, 84] ELEMENTS OF AN ANALYTIC FUNCTION 203 Let us consider now a path. L starting at the origin and ending at any point Z outside of the circle C 0? and let us imagine a moving point to describe this path, moving always in the same sense from to Z Let a^ be the point where the moving point leaves the circle ; if this point 0J were a singular point, it would be impossible to con- tinue on the path L beyond this point We shall suppose that it is not a singular point , we can then form a power series arranged in powers of or x and convergent in a cncle C x with the center a^ whose sum coincides with f(z) m the part common to the two cir- cles C and C\ To calculate /(^), /'(a? a ), we could employ, for example, an intermediate point on the radius Oa % The sum of the second series would furnish us with the analytic extension of f(z) along the path L from a v so long as the moving point does not leave the circle C t In particular, if all the path starting from ^ lies in the interior of C 19 that series will give the value of the function at the point Z If the path leaves the circle C 1 at the point or 2 , we shall form, similarly, a new power series convergent in a circle C 2 with the center # 2 , and so on* We shall suppose first that after a finite numbei of operations we arrive at a circle C p with the center a p , con- taining all the portion of the path L which follows a p9 and in partic- ular the point Z It will suffice to replace z by Z in the last series used and in those which we have obtained fiom it by term-by-term differentiation in order to find the values of /(Z), f(Z}, f'(Z\ -, with which we arrive at the point Z, that is, the final element of the function. It is clear that we arrive at any point of the path L with com- pletely determined values for the function and all its derivatives Let us note also that we could replace the circles C , C^ C 29 * -, O p by a sequence of circles similarly defined, having any points z l9 # 2 , , & q of the path L as centers, provided that the circle with the center ^ contains the portion of the path L included between z v and z i+i We can also modify the path L, keeping the same extremities, without changing the final values of /(#), /'(#), /"(), 5 for the whatever the angle a may be, for we have evidently since all the coefficients a are positive The minimum of the radius of convergence of the series (8) , when a describes the circle Cj, would then be greater than JR/2 204 ANALYTIC EXTENSION [IV, 84 FIG 33 circles C , C v - , C f cover a portion of the plane forming a kind of strip in -which the path L lies, and we can replace the path L by any other path L 1 going from 2 = to the point Z and situated in that strip Let us suppose, for definiteness, that we have to make use of three consecutive circles C , C 1? C 2 (Fig 33) Let L r be a new path lying in the strip formed by these three circles, and let us join the two points m and n If we go from to m first by the path Oa^m, then by the path Onm, it is clear that we arrive at m with the same element, since we have an analytic function in the region formed by (7 and C l Similarly, if we go from m to Z by the path ma^Z or by the path mnZ, we arrive in each case at the point Z with the same element The path L is therefore equiv- alent to the path OnmnZ, that is, to the path V The method of proof is the same, whatever may be the number of the successive cucles. In particular, we can always replace a path of any form whatever by a bioken line* 85. Singular points. If we proceed as we have just explained, it may happen that we cannot find a circle containing all that part of the path L which remains to be descubed, however far we continue the process This will be the case when the point a p is a singular point on the ciicle C p _ ly for the process will be checked just at that point If the process can be continued forevei, without arriving at a circle inclosing all that pait of the path L which lemains to be described, the points tf p _ 1? a p , a p+ i, approach a limit point X of the path Z, which may be either the point Z itself or a point lying between and Z. The point A is again a singular ijoint, and it is impossible to push the analytic extension of the unction /() along the path L beyond the point X. But if X is different from Z, it does not follow that the point Z is itself a singular point, and that we cannot go from O to Z by some other path. Let us consider, for example, either of the two functions Vl-hs and Log (1 + 2) , we could not go from * The reasoning requires a little more attention when the path L has double points, since then the strip formed hy the successive circles <7 , Ci, <7 2 , may return and cover part of itself But there is no essential difficulty IV, 85] ELEMENTS OF AN ANALYTIC FUNCTION 205 the origin to the point = 2 along the axis of reals, since we could not pass through the singular point 2 = 1 But if we cause the van- able z to describe a path not going through this point, it is clear that we shall arrive at the point z = 2 after a finite number of steps, for all the successive circles will pass through the point z = 1 It should be noticed that the preceding definition of singular points depends upon the path followed by the variable , a point X may be a singular point foi a certain path, and may not for some other, if the function has several distinct branches When two paths L 19 L{, going from the origin to Z, lead to dif- ferent elements at Z, there exists at least one singulai point in the interior of the legion which would be swept out by one of the paths, L 19 for example, if we were to deform it in a continuous manner so as to bring it into coincidence with L(, retaining always the same extremities duung the change Let us sup- pose, as is always permissible, that the two paths L 19 L{ are broken lines composed of the same number of segments Oa^q lZ and Oa&t 1{Z (Fig 34) Let a# b 2 , c 2 , . , 1 2 be the middle points of the segments a^a'^ bib'i) c i c iy * ? W? the path L^ formed by the broken line a 2 # 2 c 2 l^Z cannot be equiva- lent at the same time to the two paths L 19 L{ if it does not contain a singular point If the path Z 2 does contain a singular point, the theorem is established If the two paths L^ and L^ are not equivalent, we can deduce from them a new path 8 lying between L^ and L 2 by the same process Continuing in this way, we shall either reach a path L p containing a singular point or we shall have an infinite sequence of paths L 19 2 , 8 , . These paths will approach a limit- ing path A, for the points a l9 a a , c& 8 , approach a limit point lying between a x and a(, , and similarly foi the others This limiting path A must necessanly contain a singular point, since we can draw two paths as near as we please to A, one on each side of it, and leading to different elements for the function at Z, This could not be true if A did not contain any singular points, since the paths sufficiently close to A must lead to the same elements at Z as does A The preceding definition of singular points is purely negative and does not tell us anything about the nature of the function in 206 ANALYTIC EXTENSION [IV, 85 the neighboihood. ~No hypothesis on these singular points or on their distubution in the plane can be discarded a pnon without danger of leading to some contradiction A study of the analytic extension is required to determine all the possible cases.* 86. General problem. From what precedes, it follows that an analytic function is virtually detei mined when we know one of its elements, that is, when we know a sequence of coefficients a Q , a v a a , , a n , such that the series a Q + a^(x a)+ + a n (x <x) n H has a radius of conveigence different from zero. These coefficients being known, we are led to consider the following general problem To find the value of the function at any point p of the plane when the variable is made to describe a definitely chosen path from the point a to the point f$ We can also consider the problem of determining a prioii the singular points of the analytic function, it is also clear that the two problems are closely related to each other The method of analytic extension itself furnishes a solution of these two problems, at least theoretically, but it is piaeticable only in very particular cases For example, as nothing indicates a priori the number of intermediate series which must be employed to go from the point a to the point ft, and since we can calculate the sum of each of these series with only a certain degree of approximation, it appears impossible to obtain any idea of the final approximation which we shall reach So the investigation of simpler solutions was necessaiy, at least in particular cases Only in recent years, how- ever, has this problem been the object of thorough investigations, which have already led to some impoitant results t *Let/(a;) be a function analytic along the whole length of the segment ab of the real axis. In the neighboihood of any point <x of this segment the function can be represented by a power series whose radius of convergence J2(ar) is not zero This radius R, being a continuous function of or, has a positive minimum r Let p be a positive number less than r, and E the region of the plane swept out by a circle with, the radius p when its center describes the segment ab The function/ () is analytic In the region E and on its boundary , let M be an upper bound f 01 its absolute value , from the general formulae (14) ( 33) it follows that at any point ar of ab we have the inequality l/l<7-r (Of, I, 19T, 2d ed , 191, 1st ed ) fFor everything regarding this matter we refer the reader to Hadamard's excel- lent work, La srze de Taylor et son prolongement analytique (Naud, 1901). It con- tains a very complete bibliography. IV, 86] ELEMENTS OF AN ANALYTIC FUNCTION 207 The fact that these researches are so recent must not be attubuted entirely to the difficulty of the question, however great it may be The functions which have actually been studied successively by mathematicians have not been chosen by them aibitrarily ; rather, the study of these functions was forced upon them by the very nature of the pioblems which they encounteied Now, aside from a small numbei of transcendentals, all these functions, after the explicit elementary functions, aie defined either as the roots of equations which do not admit a foimal solution 01 as integrals of algebiaic differential equations It is clear, then, that the study of implicit functions and of functions defined by differential equations must logically have preceded the study of the geneial problem of which these two problems are essentially only very paiticular cases. It is easy to show how the study of algebraic differential equa- tions leads to the theory of analytic extension Let us consider, for concreteness, two power senes 2/(r), z(x) } arranged according to pos- itive powers of x and convergent in a cucle C of radius R descubed about the point x = as center On the other hand, let F(x y y, y r , y") , yM, z, *', , *to>) be a polynomial in x, y, y\ - , y^\ z, z',-- , z (< *\ Let us suppose that we replace y and & in this polynomial by the preceding senes, y\ y", , y^ by the successive derivatives of the series y(x), and z' } z", , z^ by the derivatives of the series #(#), the result is again a power series convergent an the circle C. If all the coefficients of that series are zero, the analytic functions y(x) and (x) satisfy, in the circle C, the relation (9) F(x,y,y', , y<*\ z, z', We are now going to prove that the functions obtained by the analytic extension of the series y(x) and z(x) satisfy the same relation in the whole of their domain of existence ** Moie precisely, if we cause the variable x to describe a path L s tax ting at the origin and proceeding fiom the circle C to reach any point a of the plane, and if it is pos- sible to continue the analytic extension of the two series y(x) and ' (cr) along the whole length of this path without meeting any singular point, the power series Y(x a) and Z(x a) with which we arrive at the point a represent, in the neighborhood of that point, two ana- lytic functions which satisfy the relation (9) For let x l be a point of the path L within the circle C and near the point where the path L leaves the circle C With the point a; 1 as center we can describe a circle C 19 partly exterior to the circle C, and there exist two power series y(x x^), z(x x^) that are convergent in the circle C l and 208 ANALYTIC EXTENSION [IV, 86 whose values are identical with, the values of the two series y (x) and z(x) in the part common to the two circles C, C : Substituting for y and s in Fthe two corresponding series, the result obtained is a power series P(x x^ eon vei gent in the circle C l Now in the part common to the two circles C, C 1 we have P(x - c a )= 0, the series P(x - xj has therefore all its coefficients zero, and the two new series y(x xj and z (x #,) satisfy the relation (9) in the circle C l Continuing in this way, we see that the relation never ceases to be satisfied by the analytic extension of the two senes y(x) and #(#), whatever the path followed by the variable may be, the proposition is thus demonstrated. The study of a function defined by a differential equation is, then, essentially only a particular case of the general problem of analytic extension But, on the other hand, it is easy to see how the knowledge of a particular relation between the analytic function and some of its derivatives may in certain cases facilitate the solution of the problem We shall have to return to this point in the study of differential equations. n NATURAL BOUNDARIES. CUTS The study of modular elliptic functions furnished Hermite the first example of an analytic function defined only in a portion of the plane We shall point out a very simple method of obtaining analytic functions having any curve whatever of the plane for a natural boundary (see 84), under certain hypotheses of a very general character concerning the curve 87. Singular lines. Natural boundaries. We shall first demonstrate a preliminary proposition * Let (j&j, a 2 , , a n , and c 1? c 2 , , c n , * be two sequences of any kind of terms, the second of which is such that $c v is absolutely convergent and has all its terms different from zero Let C be a circle with the center Z Q , containing none of the points & t in its interior and passing through a single one of these points , then the series s *POIXCARI, Acta Societatis Fenmcse, Vol XIII, 1881, GOUESAT, Bulletin ties sciences mathematiqiLes, 2d senes, Vol XI, p 109, and Vol XVII, p 247 IV, 87] NATURAL BOUNDARIES CUTS 209 represents an analytic function in the circle C which can be devel- oped in a sei les of powers of z # The circle of convergence of this set les is precisely the circle C We can cleaily suppose that Z Q = 0, for if we change & to Z Q &', a v is replaced by a v # , and c v does not change We shall also sup- pose that we have | aj = R, where R denotes the ladms of the circle C, and | Ofc| > R for i > 1 In the circle C the general teim c v /(a v z) can be developed in a power series, and that series has (\e v \/R)/(L z/R) for a dominant function, as is easily verified By a general theorem demonstrated above ( 9), the series S|<3 V | being convergent, the func- tion F(z) can be developed in a power series in the circle C, and that series can be obtained by adding term by term the power series which represent the different terms We have, then, in the circle C (10') F(z) = A Q + A l z + A z z* + + A n z + , ^JgJ^. v l^v + CO Let us choose an integer p such that V|c^| shall be smaller than v=p+l |cJ/2, which is always possible, since c 1 is not zero and since the series S|c,,| is convergent Having chosen the integer p in this way, we can write F(z) = F^z) + F z (z\ where we have set a v z v v -&-. *-/ a v z v F^(z) is a rational function which has only poles exterior to the circle C ; it is therefore developable in a power series in a circle C f with a radius R 1 > R As for F 2 (z), we have (11) JP a ()=5 + J5 1 *+ +-^+ , where 7? C l _i C P+1 i gp + 2 _|._ ^ - + We can write this coefficient again in the form but we have, by hypothesis, |a 1 / ft j< 1, and the absolute value of the sum of the series 210 ANALYTIC EXTENSION [IV, 87 is less than [flJ/2, by the method of choosing the integer p. The absolute value of the coefficient B n is theref 01 e between (cJ/2 J? n+1 and 3|c |/2 R n+l in magnitude, and the absolute value of the general teini of the series (11) lies between (|q|/2 22) \v/R\* and (3|0J/2)|*/K|, that series is theref oie divergent if \\ > R By adding to the senes F 2 (z), convergent m the circle with the radius R, a series JF^*), con- vergent in a circle of radius R 1 > R, it is clear that the sum F(z) has the circle C with the radius R for its circle of convergence ; this proves the proposition which was stated Let now L be a curve, closed or not, having at each point a definite radius of curvatuie. The series 50,, being absolutely convergent, let us suppose that the points of the sequence a v & 2 , , a t , are all on the curve L and are distributed on it in such a way that on a finite arc of this curve there are always an infinite number of points of that sequence The senes (12) *<*)-!" , z is convergent for every point not belonging to the curve Z, and represents an analytic function in the neighborhood of that point To prove this it would suffice to repeat the first part of the preced- ing pi oof, taking for the cncle C any circle with the center # and not containing any of the points a l If the cuive L is not closed, and does not have any double points, the series (12) represents an analytic function in the whole extent of the plane except for the points of the curve L. We cannot conclude from this that the curve L is a singular line; we have yet to assure ourselves that the analytic extension of F(&) is not possible across any portion of L, however small it may be. To prove this it suffices to show that the circle of convergence of the power series which represents F(z) in the neighborhood of any point 2 not on L can never inclose an arc of that curve, however small it may be Suppose that the circle C, with the center Z Q) actually incloses an arc aft of the curve L Let us take a point ar t on this arc aft, and on the normal to this arc at a % let us take a point ' so close to the point a t that the circle C t , described about the point z' as center with the radius | r a t [, shall lie entirely in the interior of C and not have any point in common with the arc aft other than the point a l itself. By the theorem which has ]ust been demonstrated, the circle C l is the circle of convergence for the power series which represents F(z) in the neighborhood of the point z r . But this is in contradiction to the general properties of power IV, 88] NATURAL BOUNDARIES CUTS 211 senes, for that circle of convergence cannot be smaller than the circle with the center z 1 which is tangent mteinally to the circle C If the curve L is closed, the series (12) represents two distinct analytic functions One of these exists only in the interior of the curve L } and for it that cuive is a natural boundary, the other function, on the contrary, exists only in the region exterior to the curve L and has the same curve as a natural boundary Thus the curve L is a natural boundary for each of these functions Given several curves, L v L^ , L p , closed or not, it will be pos- sible to form in this way series of the form (12) having these curves for natural boundaries , the sum of these series will have all these curves for natural boundaries. 88. Examples. Let AB be a segment of a straight line, and or, /3 the complex quantities representing the extremities A,B All the points 7 = (ma + np)/(m + n) , where m and n are two positive integers varying from 1 to + oo, are on the seg- ment AB, and on a finite portion of this segment there are always an infinite number of points of that kind, since the point 7 divides the segment AB in the ratio m/n On the other hand, let C m , n be the geneial term of an absolutely convergent double series The double series ma + nff *<)= represents an analytic function having the segment AB for a natural boundary We can, in fact, transform this series into a simple series with a single index xn an infinite number of ways It is clear that by adding several series of this kind it will be possible to form an analytic function having the perimeter of any given polygon as a natural boundary Another example, m which the curve L is a circle, may be defined as follows Let a. be a positive irrational number, and let v be a positive integer Let us put a = c 2 17ra , CL V = CL V = B Z I7rva . Then all the points a v are distinct and are situated on the circle C of unit radius having its center at the origin Moreover, we know that we can find two inte- gers m and n such that the difference 27r(nar m) will be less in absolute value than a number e, however small c be taken There exist, then, powers of a whose angle is as near zero as we wish, and consequently on a finite arc of the circumference there will always be an infinite number of points a v . Let us next put c v = a v /2 v , the series represents, by the general theorem, an analytic function in the circle which has the whole circumference of this cncle for a natural boundary 212 ANALYTIC EXTENSION [IV, 88 Developing each term in powers of z, we obtain for the development of F(z) the power series It is easy to prove dnectly that the function represented by this power series cannot be extended analytically beyond the circle C , for if we add to it the series for 1/(1 z), there results 2 v ' 21-z Changing in this relation s to az, then to #%, , we find the general relation (14) F(a n z) = F(z) -I j fr* * -J > which shows that the difference 2 n F(a n z) F(z) is a rational function <f> (z) hav- ing the n poles of the first order 1, I/a, - , I/a 11 - 1 . The result (14) has been established on the supposition that we have \z\ <1 and | a \ = 1 If the angle of a is commensurable with ?r, the equality (14) shows that F(z) is a rational function , to show this it would suffice to take for n an integer such that a n = 1 If the angle of a is incommensurable with TT, it is im- possible for the function F(z) to be analytic on a finite arc AB of the ciicum- ference, however small it may be. For let a-? and #-* be two points on the arc AB(n>p) The numbers n and $ having been chosen in this way, let us suppose that z is made to approach arf , OPZ will approach a*--?, and the two functions F(z) and F(a n z) would approach finite limits if F(z) were analytic on the arc AB. Now the relation (14) shows that this is impossible, since the function <j> (z) has the pole or P. An analogous method is applicable, as Hadamard has shown, to the series considered by Weierstrass, (15) where a is a positive integer > 1 and 6 is a constant whose absolute value is less than one This series is convergent if | z \ is not greater than unity, and diver- gent if | | is greater than unity The circle C with a unri radius is therefore the circle of convergence. The circumference is a natural boundary for the func- tion F(z) IPor suppose that there are no singular points of the function on a finite arc ap of the circumference If we replace the variable & in F(z) by 2C 2 * W/C *, where k and h are two positive integers and c a divisor of a, all the terms of the series (15) after the term of the rank h are unchanged, and the difference F(z) F(z&*-'<*) is a polynomial Neither would the function F(z) have any singular points on the arc ojft., which is denved from the arc ocp by a rotation through an angle 2 kir/c h around the origin Let us take h large enough to make 2 ?r/c* smaller than the arc aft , taking successively k = 1, 2, - * , c*, it is clear that the arcs o^, <*a/3 2 , . . cover the circumference completely The IV, 89] NATURAL BOUNDABIES CUTS 213 function F(z) would therefore not have any smgulai points on the circumfer- ence, which is absurd ( 84) This example presents an interesting peculiarity, the series (15) is absolutely and uniformly convergent along the circumference of C It represents, then, a continuous function of the angle along this circle * 89 Singularities of analytical expressions. Every analytical expres- sion (such as a series whose different terms are functions of a vari- able #, or a definite integral in which that variable appears as a parameter) represents, under certain conditions, an analytic function in the neighborhood of each of the values of z for which it has a meaning If the set of these values of z covers completely a connected region A of the plane, the expiession considered represents an analytic function of z in that region A , but if the set of these values of z forms two or more distinct and separated regions, it may happen that the analytical expression considered represents entirely distinct functions in these different regions We have already met an exam- ple of this in 38 There we saw how we could form a series of rational terms, convergent in two curvilinear triangles PQR, P'Q'R* (Fig. 16), whose value is equal to a given analytic function f(z) in the triangle PQR and to zero in the triangle P'Q'R* By adding two such series we shall obtain a series of rational terms whose value is equal to f(z) in the triangle PQR and to another analytic function <f) (z) in the triangle P'Q'R'. These two functions f(z) and <j> (z) being * Fredholm has shown, similarly, that the function represented by the series where a is a positive quantity less than one, cannot be extended beyond the circle of convergence (Comptes rendus, March 24, 1890) This example leads to a result which is worthy of mention On the circle of unit radius the senes is conveigent and the value F(0) S o[cos (r$0) + 1 sm (n 2 0)] is a continuous function of the angle 9 which has an infinite number of derivatives This function F(9) cannot, however, be developed in a Taylor's series in any interval, however small it may be Suppose that in the interval (#o - a, 0o + a) we actually have The series on the right represents an analytic function of the complex variable 6 in the circle c with the radius a described with the point Q for center To this circle c corresponds, by means of the relation z- e fl , a closed region A of the plane of the vari- able z containing the arc 7 of the unit circle extending from the point with the angle - <* to the point with the angle + <* Tnere would exist, then, in this region A an analytic function of z coinciding with the value of the series Sa^ 8 along 7 and also m the part of A within the unit circle, this is impossible, since we cannot extend the sum of the senes beyond the circle 214 ANALYTIC EXTENSION [IV, 89 arbitiary, it is clear that the value of the series in the triangle P'Q'R' will in general bear no i elation to the analytic extension of the value of that series in the triangle PQR The following is anothei veiy simple example, analogous to an example pointed out by Schioder and by Tanneiy The expression (1 _- s R )/(l + s n ), where n is a positive integer which increases in- definitely, approaches the limit +1 if |*|< 1, and the limit 1 if |*| > 1 If |*| = 1, this expiession has no limit except foi z = l Now the sum of the first n terms of the series is equal to the piecedmg expression This series is therefore conver- gent if || is different from unity Hence it repiesents + 1 in the interior of the ciicle C with the radius unity about the origin as centei, and 1 at all points outside of this circle. Now let/(s), $(z) be any two analytic functions whatever, for example, two integial functions Then the expiession 1 '2 J is equal to f(z) in the interior of C, and to < (z) in the region ex- tenor to C. The circumference itself is a cut for that expression, but of a quite different natuie from the natural boundaries which we have just mentioned. The function which is equal to $(&) in the interior of C can be extended analytically beyond C , and, similarly, the function which is equal to \l/(&) outside of C can be extended analytically into the interior. Analogous singularities present themselves in the case of functions represented by definite integrals The simplest example is furnished by Cauchy's integral, if/() is a function analytic within a closed curve F and also on that curve itself, the integral /_ v -} C*- V2Wj r *-x represents f(x) if the point x is in the interior of T The same inte- gral is zero if the point x is outside of the curve T, for the function /()/( x) is then analytic inside of the curve Here again the curve T is not a natural boundary for the definite integral. Similarly, the definite integral ^ 2ff ctn [(* se)/2]<fc has the real axis as a cut ; it is equal to + 2 iri or 2 TTI, according as x is above or below that cut ( 45). IV, 90] NATURAL BOUNDARIES CUTS 215 90 Semite's formula An interesting result due to Hermite can be brought into i elation with the preceding discussion * Let F(t, z), G (t, z) be two analytic functions of each of the variables t and z , for example, two polynomials or two power &enes convergent foi all the values of these two vanables Then the definite integral "" taken over the segment of a straight line which joins the two points a and j3, represents, as we shall see later ( 95), an analytic function of z except for the values of z which are roots of the equation G- (t, z) = 0, wheie t is the complex quantity corresponding to a point on the segment a/3 This equation theiefoie determines a finite or an infinite number of cuives foi which the mtegial $(z) ceases to have a meaning Let AB be one of these curves not having any double points In older to consider a veiy precise case, we shall suppose that when t de&ciibes the segment #, one of the roots of the equation G(t, 2) = describes the aic AB, and that all the other roots of the same equation, if there are any, remain outside of a suitably chosen closed curve sui rounding the arc AB, so that the segment <ar/3 and the aic AB coirespond to each other point to point The integral (16) has no meaning when z falls upon the aic AB , we wish to calculate the difference between the values of the function $ (z) at two points JT, N'i lying on opposite sides of the arc AB, whose distances from a fixed point M of the arc AB are infinitesimal Let & + <?, + <' be the thiee values of z corresponding to the thiee points Jf, N, N' respectively To these thiee points coirespond in the plane of the vanable t, by means of the equation G (, z) = 0, the point m on <arj3, and the two points n, n" on opposite sides of a$ at infinitesimal distances fiom m Let 0, 6 + v, Q + if be the coi- le&pondmg values of t In the neighborhood of the segment aft let us take a point 7 so near aft that the equation G (, + c) = has no other root than t = Q H- 1? in the interior of the triangle afiy (Fig 35) The function Ffa f + )/<? (^ ^ 4. e ) of the variable t has but a single pole 6 + ijm the inteuor of the triangle or/37, and, according to the hypotheses made above, this pole is a simple pole Applying Cauchy's theorem, we have, then, the relation (17) T The two integrals f, f y " are of the same form as $(2) , they represent re- spectively two functions, ^(z), < 2 (2), which are analytic so long as the variable is not situated upon certain curves Let AC and BC be the curves which cor- respond to the two segments #7 and fty of the t plane, and which are at infinitesimal distances from the cut AB associated with < (z) Let us now give * HERMITE, Sur quelqites points de la theorie des fonetions (Crelle's Journal, Vol XCI) 216 ANALYTIC EXTENSION [IV, 90 the value f -f e' to z , the coriespondmg value of t is 6 + 17', represented by the point TI', and the function F(t, f + J)/G(t, + O of * 1S analytic m the interior of the triangle apy We have, then, the relation G+) * ffftr+O y <*,*+ g the two formulse (17) and (18) term by term, we can as follows <*,*+ subtracting the two formulse (17) and (18) term by term, we can wnte the result But since neithei of the functions ^(z), $ 2 (z) has the line AB as a cut, they are analytic m the neighboihoocl of the point z = f, and by making e and e' ap- proach zeio we obtain at the limit the difference of the values of $(z) in two points infinitely near each other on opposite sides ol AB We shall wnte the result in the abridged form (19) *w-*w = toSt&, 30 this is Hermite's formula It is seen that it is very simply related to Cauchy's theorem * The demonstration indicates clearly how we must take the points N and N' , the point N(f + e) must be such that an observer descubmg the segment a$ has the corresponding point 9 + 17 on his left It is to be noticed that the arc AB is not a natural boundary for the function $(z). In the neighborhood of the point JV' we can replace *(z) by [$!() + $ 2 ( z )] according to the relation (18) Now the sum ^(z) H- $ 2 ( z ) 1S an analytic function in the curvilinear triangle A CB and on the arc A B itself, as well as in the neighborhood of N'. Theiefore we can make the variable z cross the arc AB at any one of its points except the extremities A and B without meeting any obstacle to the analytic extension. The same thing would be true if we were to make the variable z cross the arc AB in the opposite sense Example. Let us consider the integial where the Integral is to be taken over a segment AB of the real axis, and where f(t) denotes an analytic function along that segment AB Let us represent z on the same plane as t . The function $ (z) is an analytic function of z in the neighborhood of every point not located on the segment AB itself, which is a cut for the integral The difference * (N) * (N*) is here equal to 2 m/(fl, where f is a point of the segment AB When the variable z crosses the line A B, the analytic extension of $ (z) is represented by * (z) 2 mf(z) This example gives rise to an important observation The function $ (z) is still an analytic function of z, even when/(4) is not an analytic function of , provided that f(t) is continuous between a and p ( 33) But m this case the preceding reasoning no longer applies, and the segment AB is in general a natural boundary for the function * GOURSA.T, Sur un tMortme de M Herrmte (Acta mathematica, Vol I) IV, Exs ] EXERCISES 217 EXERCISES 1. Pmd the lmes.of discontinuity for the definite integrals taken along the stiaight line which joins the points (0, 1) and (ot, b) respec- tively , determine the value of these integrals f 01 a point z not located on these boundaries 2 Consider f om circles with radii 1/V2, having for centers the points + 1, -f i, 1, i The region exterior to these foui circles is composed of a finite region A^ containing the origin, and of an infinite region A 2 Construct, by the method of 38, a series of rational functions which converge in these regions, and whose value in A l is equal to 1 and in A 2 to Verify the result by finding the sum of the series obtained 3 Treat the same questions, considenng the two regions interior to the circle of radius 2 with the center for origin, and exterior to the two circles of radius 1 with centers at the points + 1 and 1 respectively [APPELL, Acta mathematica, Vol. I.] 4 The definite integral fame taken along the real axis, has for cuts the straight lines x = (2 k -f 1) ?r, where k is an integer Let = (2 k + 1) TT + i be a point on one of these cuts The dif- feience in the values of the integial in two points infinitely close to that point on each side of the cut is equal to ie (erf + e-). [HERMITE, Crelle's Journal, Vol XCI ] 5. The two definite integrals j C Ld, J a = C *^-i dt. J-o, Z J_ * taken along the leal axis, have the axis of reals for a cut in the plane of the variable z Above the axis we have J 2 TTI, J 9 = 0, and below we have J = 0, J Q s= 2m From these lesults deduce the values of the definite integrals + a* + 00 Q It '***** L, ^j=-. *- - - [HERMITE, CreZZe's Journal, Vol XCI ] 6 Establish by means of cuts the formula (Chap. II, Ex 15) -.+ git ir 'Consider the integral "*" - " fl+-a ri -oo 1+e* [EERMITE, Cretins Journal, Vol XCI,] 218 ANALYTIC EXTENSION [IV, Exs which has all the stiaight lines y = (2 k -f 1) TT f or cuts, and which remains con- stant in the strip included between two consecutive cuts Then establish the relations where 2; and z 4- 2 TTI are two points separated by the cut y TT ) 7*. Let/(2) be an analytic function in the neighborhood of the origin, so that f(z) =Sa n z n Denote by F(z) =SOn2 n / n ' the associated integral function It is easily proved that we have (1) ^ ' where the integral is taken along a closed curve C, including the origin within it, inside of which f(z) is analytic From this it follows that r l _, , 1 rf(u) (2) / e-*F(az)da = - \ J -^- x ' Jo 2m Jc u where I denotes a real and positive number If the leal part ot z/u remains less than 1 e (where e> 0) when u describes the curve (7, the integral C l I Jo e*V Vda o approaches u/(u s) uniformly as Z becomes infinite, and the formula (2) be- comes at the limit rv^(a*x*a=-i- c m^ = Jo v ' Zm J?) u z This result is applicable to all the points within the negative pedal curve of C [BOREL, Lemons sur les series divergences ] 8*. Let/(2) = SOn2 n , ^> (2;) = S&nZ" be two power series whose radii of conver- gence are r and p respectively The series has a radius of convergence at least equal to rp, and the function ^ (z) has no other singular points than those which are obtained by multiplying the quanti- ties coriesponding to the different singular points off(z) by those corresponding to the singular points of $ (z) [HADAMABD, Ada mathematica, Vol. XXIII, p 55 ] CHAPTER V ANALYTIC FUNCTIONS OF SEVERAL VARIABLES I GENERAL PROPERTIES In this chapter we shall discuss analytic functions of several independent complex vanables For simplicity, we shall suppose that there aie two variables only, but it is easy to extend the results to functions of any number of variables whatever 91 Definitions Let & = u -f- vi, l = w -f ti be two independent complex variables , every other complex quantity Z whose value depends upon the values of & and & 1 can be said to be a function of the two variables & and #' Let us represent the values of these two variables & and 2' by the two points with the coordinates (u, v) and (w, t) in two systems of rectangular axes situated in two planes P, P', and let A, A 1 be any two portions of these two planes We shall say that a function Z =/(#, #') is analytw in the two regions A, A' if to every system of two points , ', taken respectively in the regions A t A', corresponds a definite value of f(z, #'), varying continuously with & and 2', and if each of the quotients /(* + *,)-/(*,') /(*,*' + &)~/0,* f ) h ' k approaches a definite limit when, and #' remaining fixed, the absolute values of h and k approach zero These limits are the partial derivatives of the function /(#, '), and they are represented by the same notation as in the case of real variables Let us separate in /(#, #') the real part and the coefficient of i } f(z y #') = X -f Yi ; X and Y are real functions of the four independ- ent real variables u, v, w, t, satisfying the four relations &*T__aF 2____F ? = ?Z = _Z 0tt ~~ dv ' 8v ~ du,' dw ~~ dt ' fa ~~ dw' the significance of which is evident.* We can eliminate T in six * If z and tf are analytic functions of another variable a:, these relations enable us to demonstrate easily that the derivative of /(z, "wrtk respect to as is obtained by the usual rule which gives the derivative of a function of other functions The formulae of the differential calculus, in particular those for the change of variables, apply, therefore, to analytic functions of complex variables 219 220 SEVERAL VARIABLES [V,91 different ways by passing to derivatives of the second order, but the six relations thus obtained i educe to only four (J^^JjL^O d * X &X = Q dudt dvdw ' dudw dvdt ' d*X PX _ VX d^X Jtf + ~W " ' dw* + 9f ~~ Up to the present time little use has been made of these relations for the study of analytic functions of two vanables One reason for this is that they are too numeious to be convenient 92. Associated circles of convergence. The properties of power series in two real variables (I, 190-192, 2d ed. ; 185-186, 1st ed ) are easily extended to the case where the coefficients and the variables have complex values Let (2) F(z,z') be a double series with coefficients of any kind, and let We have seen (I, 190, 2d ed) that theie exist, in general, an infinite number of systems of two positive numbers R 9 R ! such that the series of absolute values (3) 2A mn ZZ' n is convergent if we have at the same time Z<R and Z'<R', and divergent if we have Z>R and Z'>R' Let C be the cucle de- scribed in the plane of the variable # about the origin as center with the radius R ; similarly, let C 1 be the circle described in the plane of the variable #' about the point #' = as center with the radius R 1 (Fig. 36) The double series (2) is absolutely con vei gent when the variables s and #' are respectively in the interior of the two cucles C and C', and divergent when these variables are respectively extenor to these two circles (I, 191, 2d ed ; 185, 1st ed ) The circles C, C" are said to form a system of associated circles of convergence This set of two circles plays the same part as the circle of convergence for a power series in one variable, but in place of a single circle there is an infinite number of systems of associated circles for a power series in two variables. Tor example, the series ml n\ v, GENERAL PROPERTIES 221 is absolutely convergent if |g|-f |2 r |<l, and in that case only Every pair of circles C, C" whose radii R, R' satisfy the relation R -f R* = 1 is a system of associated circles. It may happen that we can limit ourselves to the consideration of a single system of asso- ciated circles , thus, the series S2 m 2'* is convergent only if we have at the same time \z\ < 1 and |#'| < 1 Let C l be a circle of ladius R % <R concentric with C; similarly, let Ci be a circle of radius Ri<R' concentric with C", when the variables 2 and 2' remain within the circles C^ and C[ respectively, the series (2) is uniformly convergent (see I, 191, 2d ed , 185, 1st ed ) and the sum of the series is therefore a continuous function F(z, 2') of the two variables 2, 2' in the interioi of the two circles C and C". Differentiating the series (2) term by term with respect to the variable 2, for example, the new series obtained, ^ma mn z m " 1 z' n } is again absolutely convergent when 2 and 2' lemain in the two circles C and C" respectively, and its sum is the denvative dF/dz of F(z, 2') with respect to 2 The proof is similar in all respects to the one which has been given for real variables (I, 191, 2d ed , 185, 1st ed.) Simi- larly, jF(2, 2') has a partial derivative dF/dz' with respect to 2', which is represented by the double series obtained by differentiating the series (2) term by term with respect to 2 r The function F(z, 2 r ) is therefore an analytic function of the two variables 2, # f in the pre- ceding region The same thing is evidently true of the two deriva- tives 8F/dz, dF/8z', and therefore F(z, 2^ can be differentiated term 222 SEVERAL VARIABLES [V, 92 by term any number of times ; all its partial derivatives are also analytic functions Let us take any point of absolute value r in the interior of C, and from this point as center let us describe a circle c with ladius R r tangent internally to the circle C In the same way let z' be any point of absolute value r 1 < It', and c' the circle with the point z 1 as center and R 1 r 1 as ladius Finally, let z + h and z' 4- k be any two points taken in the cncles c and c' respectively, so that we have |*| + |A|<jR, \z'\ + \k\<R' If we replace z and z 1 in the series (2) by z + h and z 1 + k, we can develop each term in a senes proceeding accoiding to poweis of h and k, and the multiple senes thus obtained is absolutely con vei gent Arranging the senes according to powers of h and k, we obtain the Taylor expansion (*) 93 Double integrals. When we undertake to extend to functions of several complex variables the general theorems which Cauchy deduced from the consideration of definite integrals taken between imaginary limits, we encounter difficulties which have been com- pletely elucidated by Poincare * We shall study here only a very w FIG 37 simple particular case, which will, however, suffice for our subse- quent developments. Let /(, *) be an analytic function when the variables #, z 1 remain within the two regions A } A' respectively Let us consider a curve ah lying in A (Fig 37) and a curve a!V in A\ and let us divide each of these curves into smaller arcs by any number of points of division. Let , #1, 2 , , ^-i? ***> > z * PoiffCAB6, Sur ks res^dus ties integrates doubles (Acta mattemahca, Vol IX), V,93] GENERAL PROPERTIES 223 be the points of division of ab, where Z Q and Z coincide with a and b, and let z' Q , &{, z' 2 , , ' h _ l9 z' h , , *m-u z * ^ e tne points of division of a'b', where J and Z 1 coincide with a' and V The sum (5) s=j? V /( St _ u 4.0 (* - % _ x ) - ,,'.0, 11 A=l taken with respect to the two indices, approaches a limit, when the two numbers m and n become infinite, 111 such a way that the abso- lute values \z k ^_ T | and |i 4-i| appioach zero. Let f(z, ') = X+Yi, wheie X and Y aie real functions of the four vanables u, v, w, t, and let us put k = % -j- # A fc, *j[ = u\ + t h i The general term of the sum S can be wntten in the form Xi( k _ ly v k _^ w h _ l} t h + X [% %-i 4- t(^ t *JL- and if we cany out the indicated multiplication, we have eight partial pioducts Let us show, for example, that the sum of the partial pioducts, approaches a limit We shall suppose, as is the case in the figure, that the curve ab is met in only one point by a parallel to the axis Ov, and, siniilaily, that a parallel to the axis Ot meets the curve a'b* in at most one point Let v = <(w), t = ty(w) be the equations of these two curves, U Q and U the limits between which u vanes, and W Q and W the limits between which w varies. If we replace the variables v and t in X by <(w) and $(w) respectively, it becomes a continuous function P(u, w) of the vanables u and w, and the sum (6) can again be wntten in the fonn n m (6 f ) As m and TI become infinite, this sum has for its limit the double integral ffPfyt, w)dudw extended over the rectangle bounded by the straight lines u = , u = C7", w = w , w = W. This double integral can also be expressed in the form /> u /*w I du I P(u y Ju n Jw n 224 SEVERAL VARIABLES [V,93 or again, by introducing line integrals, in the form (7) C du I X(u, v, w, t)dw. J(ab) J(a'b') In this last expression we suppose that u and v are the coordinates of any point of the arc ab, and w y t the coordinates of any point of the arc a'b'. The point (u, v) being supposed fixed, the point (w, t) is made to descube the aic a'b', and the line integral fXdw is taken along a'b'. The result is a function of w, v, say R (u, v) , we then calculate the line integral/is (u, v) du along the arc ab The last expiession (7) obtained for the limit of the sum (6) is applicable whatever may be the paths ab and a'b'. It suffices to break up the arcs ab and a'b' (as we have done repeatedly before) into arcs small enough to satisfy the previous requirements, to associate in all possible ways a portion of ab with a portion of a'b', and then to add the results Proceeding in this way with all the sums of par- tial products similar to the sum (6), we see that S has for its limit the sum of eight double integrals analogous to the integral (7) Representing that limit byJJjF(#, z'^dzdz', we have the equality ')dzdz'= i du C Xdw \ dv \ Xdt J(ab) J(a'b') /(&) /(a'&') - C du C Ydt - i dv C Ydw J(ab) */('&') c/(a&) t/(a'&') + i C du C Ydw ~i I dv C Ydt J(a&) J(a'b') /(a&) */(a'&') + i I du I Xdt +i I dv I Xdw, /(<*&) t/(a'&') */(&) J(a'b') which can be written in an abridged form, ffp(z, z 1 ) dzdz' = r (du + idv) C (X + <F) (dw + idf), JJ */(&) c/O'&') (8) /(&) or, again, (9) ffffa z^dzdz' = C ds C F(z> s') dd JJ t/(o&) J(a'b') The formula (9) is precisely similar to the formula for calculating an ordinary double integral taken over the area of a rectangle by means of two successive quadratures (I, 120, 2d ed. ; 123, 1st ed ). We calculate first the integral fF(z, z'} dz' along the arc a'b', supposing V,94] GENERAL PROPERTIES 225 2 constant, the result is a function $(2) of 2, which, we integrate next along the arc db As the two paths db and a)V enter in exactly the same way, it is clear that we can interchange the order of integrations Let M be a positive number greater than the absolute value of F(z, z 1 ) when & and z 1 descube the arcs db and a)V If L and L' denote the lengths of the respective arcs, the absolute value of the double integral is less than MIL' ( 25) When one of the paths, a'b' for example, forms a closed curve, the integral f^ a > bf) F(z y z')dz' will be zero if the function F(z, z'*) is analytic for all the values of z' in the interior of that cuive and for the values of z on db. The same thing will then be true of the double integral 94. Extension of Cauchy's theorems. Let C, C 1 be two closed curves without double points, lying respectively in the planes of the variables z and z 1 , and let F(z, z') be a function that is analytic when z and z* remain in the regions limited by these two curves or on the curves themselves Let us consider the double integral I=| dz I ^ 9 *'\ r-j J(O J(co v 5 ~~ * A* ~~ ^ / where a; is a point inside of the boundary (7 and where x 1 is a point inside of the boundary C ! , and let us suppose that these two bound- aries are described in the positive sense The integral JP(g, z')dz' where & denotes a fixed point of the boundary (7, is equal to 2 iri F(z, d)/( x) We have, then, or, applying Cauchy's theorem once more, /=: 4-77^(0;, a? f ) This leads us to the formula (10) ^ ' is completely analogous to Caueliy's fundamental formula, and from which we can derive similar conclusions Erom it -we deduce 226 SEVERAL VARIABLES [V, 94 the existence of the partial derivatives of all orders of the function F(z, s') m the regions considered, the derivative d m+n F/dx m dx fn hav- ing a value given by the expiession S M+n F m* In order to obtain Taylor's formula, let us suppose that the boundanes C and C' are the circumferences of cncles Let a be the center of C, and R its radius, b the center of C", and R' its radius The points x and x 1 being taken respectively in the interior of these circles, we have \x a\ = r < R and \x' b \ = r 1 < R' Hence the rational fraction (* - oj)(*' -a; 1 ) [*--( a)][*' - can be developed in powers of x a and x 1 b, where the series on the right is uniformly conveigent when 2 and z f describe the circles C and C' respectively, since the absolute value of the general term is (r/R) m (r'/R') n /RR f We can theiefore replace !/( )(' x') by the preceding series in the relation (10) and integrate term by term, which gives Making use of the results obtained by replacing x and x' by a and b in the relations (10) and (11), we obtain Taylor's expansion in the form where the combination m = 71 = is excluded from the summation JVbte. The coefficient a^ of (as a) m (x' 5) n in the preceding series is equal to the double integral '_ 7>Y + i V,95] GENERAL PROPERTIES 227 If M is an upper bound foi | F(z, ')| along the cncles C and C.', we have, by a previous general remark, The function M is therefore a dominant function for F(x } #') (I, 192, 2d ed ? 186, 1st ed ) 95. Functions represented by definite integrals. In order to study certain functions, we often seek to express them as definite integrals in which the independent variable appears as a parameter under the integral sign We have already given sufficient conditions under which the usual rules of differentiation may be applied when the variables are real (I, 98, 100, 2d ed , 97, 1st ed.). We shall now reconsider the question for complex variables Let F(z, #') be an analytic function of the two variables & and & f when these variables remain within the two regions A and A 1 respec- tively. Let us take a definite path L of finite length in the region A, and let us consider the definite integral (13) *(*)= f F(z,x)dz, J(Ly where x is any point of the region A 1 . To prove that this function < (x) is an analytic function of x 9 let us describe about the point x as center a circle C with radius R, lying entirely in the region A 1 Since the function F(z, '} is analytic, Cauchy's fundamental formula gives whence the integral (13) can be written in the form 1 ' x Let x + Ax be a point near x in the circle C ; we have, similarly,, F(z,z')dz' 228 SEVERAL VARIABLES [V, 95 and consequently, by repeating the calculation already made ( 33), F (*, A* - 2** JnJto (*'-*? Aar F(, Let M be a positive number greater than the absolute value of F(, &') when the variables and s* describe the curves L and C respectively , let S be the length of the cui ve L , and let p denote the absolute value of Ax. The absolute value of the second integral is less than M hence it approaches zero when the point a + Ax approaches x in- definitely It follows that the function < (a?) has a unique derivative which is given by the expiession _. ,, , i r 7 r F (*, *'(7!)=- - I dz \ -j-f V ' 2 ^J(L) J(P) (*' But we have also ( 33) and the preceding relation can be again wiitten *'()= I ^-dz. Thus we obtaui again the usual formula for differentiation under the integral sign The reasoning is no longer valid if the path of integration L extends to infinity. Let us suppose, for definiteness, that i is a ray proceeding from a point a Q and making an angle & with the real axis. We shall say that the integral $(x)= I F(z, x)dz is uniformly convergent if to every positive number e there can be made to correspond a positive number N such that we have JF(,aj)<b V,96] GENERAL PROPERTIES 229 provided that p is greater than N 9 wherever x may be in A 1 By dividing the path of integration into an infinite number of recti- lineal segments we piove that every uniformly convergent integial is equal to the value of a uniformly convergent series whose teims are the integrals along certain segments of the infinite lay L. All these integrals are analytic functions of x, therefore the same is tiue of the integral f~F(z, x)d* ( 39) It is seen, in the same way, that the ordinary formula for differen- tiation can be applied, provided the integral obtained, ^(dF/dx)dz, is itself uniformly convergent. If the function F(z, 2') becomes infinite for a limit # of the path of integration, we shall also say that the integral is uniformly con- vergent in a certain region if to every positive number e a point a Q + TI on the line L can be made to correspond in such a way that I/' *,.> I */a + Tj where 5 is any point of the path L lying between a Q and a Q -f 77, the inequality holding for all values of x in the region considered. The conclusions aie the same as in the case where one of the limits of the integral is moved off to infinity, and they aie established in the same way 96 Application to the T function The definite integral taken along the real axis (15) T(2) which we have studied only for real and positive values of z (I, 94, 2d ed ; 92, 1st ed ), has a finite value, provided the real part of z, which we will denote by *R(), is positive. In fact, let z = x + yi, this gives \t 9t - 1 Since the integral +CD i t JQ has a finite value if x is positive, it is clear that the same is true of the integral (15) (I, 91, 92, 2d ed , 90, 91, 1st ed.) This integral is uniformly con- vergent m the whole region defined by the conditions N> < R(z)>i], where N and 7j are two arbitrary positive numbers In fact, we can write r(z)= f V- /0 and it suffices to prove that each of these integrals on the right is uniformly convergent Let us prove this for the second integral, for example. Let I be a positive number greater than one If *% (z) < N, we have ir*- 230 SEVERAL VARIABLES [V, 96 and a positive number A can be found laige enough to make the last integral less than any positive number e whenevei I ^ A The function r (z), defined by the integral (15), is theiefoie an analytic function m the whole legion of the plane lying to the light of the ^-axis This function r (z) satisfies again the relation (16) obtained by mtegiation by parts, and consequently the more general relation (17) r(z + n) = z(z + l) (z + n-l)r(z), which is an immediate consequence of the other This piopeity enables us to extend the definition of the T function to values of z whose real part is negative For consider the function where n is a positive integer The numeiator r (z + n) is an analytic function of z defined for values of z for which ft (z) > n , hence the function ^ (z) is a function analytic except for poles, defined for all the values of the variable whose real part is greater than n Now this function ^ (z) coincides with the analytic function r (z) to the light of the y-axis, by the relation (17), hence it is identical with the analytic extension of the analytic function F (z) in the strip included between the two stiaight lines ^(z) = 0, ft(z) = n Since the number n is arbitraiy, we may conclude that there exists a function which is analytic except for the poles of the first order at the points z = 0, z = 1, z = 2, -, z = n, - , and which is equal to the integral (15) at all points to the right of the y-axis This function, which is analytic except for poles in the finite plane, is again represented by r (2) , but the formula (15) enables us to compute its numerical value only if we have ^ (z) > If ^ (z) < 0, we must also make use of the i elation (17) in order to obtain the numerical value of that function We shall now give an expression for the r function which is valid for all values of z Let S(z) be the integral function S(z) = zl n- which has the poles of r (z) for zeros The product 8 (z) T (z) must then be an integral function. It can be shown that this integral function is equal to 6-0% where <7 is Euler's constant* (I, 18, Ex., 2ded , 49, Note, 1st ed ), and we denve from it the result (19) which shows that 1/T (z + 1) is a transcendental integral function. HBEMIMI, Cours ^Analyse, 4th ed , p 142 V,97] GENERAL PROPERTIES 231 97, Analytic extension of a function of two vanables. Let u = F(z, tf) be an analytic function of the two variables z and z f when these two variables remain respectively in two connected regions A and A! of the two planes in which we lepresent them It is shown, as m the case of a single variable ( 83), that the value of this function foi any pair of points 2, z' taken in the regions A, A' is detei mined if we know the values of F and of all its partial derivatives for a pair of points z = a, z f = o taken in the same regions It now appears easy to extend the notion of analytic extension to functions of two complex variables Let us consider a double series Sa mn such that there exist two positive numbers 7, having the following property the series (20) F(z,z^) = ^a mn z^^ is convergent if we have at the same time |s| < r, |z' | < r', and divergent if we have at the same time \z\ > r, \i\ > r' The preceding series defines, then, a function F(z, z') which is analytic when the vanables z, zf remain lespectively in the circles 0, C' of radii r and K , but it does not tell us anything about the natuie of this function when we have \z\>r or \z f \>r" Let us suppose for defimteness that we cause the variable z to move over a path L from the origin to a point Z exterior to the circle C, and the variable f to travel over another path L' from the point z' = to a point Z' exterior to the circle C" Let a and ft be two points taken respectively on the two paths and I/, a being in the intenor of C and ft m the interior of C' The series (20) and those which are obtained from it by successive differentiations enable us to form a new power senes, (21) which is absolutely convergent if we have \z a \ < r t and [zf ft \ < r^ where r x and r{ are two suitably chosen positive numbeis Let us call C l the circle of radius r t described about the point a as center in the plane of z, and C{ the circle of radius r{ described in the plane of z f about the point ft as center If z is in the part common to the two circles and C 1? and the point z* m the part common to the two circles C' and C^ the value of the series (21) is the same as the value of the series (20) If it is possible to choose the two numbers r t and r{ in such a way that the circle (7, will be partly exterior to the circle (7, or the circle G{ partly exterior to the circle C', we shall have extended the definition of the function F(z, z') to a region extending beyond the first Continuing in this manner, it is easy to see how the function F(z, zf) may be extended step by step. But there appears here an important new consideration : It is necessary to take into account the way in which the variables move unth respect to each other on their respective paths The following is a very simple example of this, due to Sauvage * Let u = "Vz zf 4- 1 , for the initial values let us take z=:z' = Q,u=:l, and let the paths described by the variables 2, z* be defined as follows . 1) The path described by the variable z f is composed of the rectilinear segment from the origin to the point z* = 1 2) The path described by z is composed of three semi circumferences the first, OM A (Fig. 38), has its center on the real a:ns to * Premiers pnneipes de la the'one generate tie* fonctwns de plusieurs variables (Annales de la Faculte" ties Sciences de Marseille, Vol. XIV) This memoir is an excellent introduction to the study of analytic functions of seveial variables 232 SEVERAL VARIABLES [V, 97 the left of the origin and a ladius less than 1/2 , the second, ANB, also has its center on the real axis and is so placed that the point 1 is on its diameter AB , finally, the third, BPC, has for its centei the middle point of the segment joining the point B to the point C(z = 1) The fiist and the thud of these seinicircum- ferences are above the leal axis, and the second is below, bo that the bound- ary OMANBPCO incloses the point z = 1 Let us now select the following movements 1) tf remains zero, and z descubes the entiie path OABC , 2) z lemams equal to 1, and z f descubes its whole path If we consider the auxihaiy vanable t = z z', it is easily seen that the path described "by the variable , when that variable is represented by a point on the . 38 2 plane, is precisely the closed boundary OABCO which surrounds the critical point t = 1 of the radical V$ + 1. The final value of u is therefore u = 1 On the other hand, let us select the following procedure 1) z remains zero and z' varies from to 1 e (e being a very small positive number) ; 2) z' remains equal to 1 e, and z describes the path OABC , 3) z remains equal to 1, and z* varies from 1 e to 1 When zf varies from to 1 e, the auxiliary variable t descubes a path 00' ending in a point 0' very near the point 1 on the real axis When z describes next the path OABC, t moves over a path (/A'tfO' congruent to the preceding and ending in the point C' '(OC' = e) on the real axis Finally, when sf varies from 1 e to 1, t passes from C" to the origin. Thus the auxiliary variable t describes the closed boundary OO'A'B'C'O which leaves the point 1 on its exterior, provided e is taken small enough. The final value of u will therefore be equal to + 1. Very much less is known about the nature of the singularities of analytic functions of several variables than about those of functions of a single variable One of the greatest difficulties of the problem lies m the fact that the pairs of singular values are not isolated * *For everything regarding this matter see a memoir by Pomcarl in the Acta mathematica (Vol X2CVT), and P Cousin's thesis (Ibid Vol XIX) V,98] IMPLICIT FUNCTIONS 233 n IMPLICIT FUNCTIONS ALGEBRAIC FUNCTIONS 98. Weierstrass's theorem. We have already established (I, 193, 2d ed , 187, 1st ed ) the existence of implicit functions defined by equations in which the left-hand side can be developed m a power series proceeding in positive and increasing powers of the two variables The arguments which were made supposing the variables and coefficients real apply without modification when the variables and the coefficients have any values, real or imaginary, provided we retain the other hypotheses We shall establish now a more general theorem, and we shall preserve the notations previously used in that study. The complex variables will be denoted by x and y. Let F(x, y) be an analytic function in the neighborhood of a pair of values x = a, y = /3, and such that we have F(a, /?) = 0. We shall suppose that a = ft 0, which is always permissible The equation F(Q, ?/) = has the root y = to a certain degree of mul- tiplicity. The case which we have studied is that in which y = is a simple root , we shall now study the geneial case where y = is a multiple root of order n of the equation .F(0, y) = 0. If we ai range the development of F(x, y) in the neighborhood of the point x = y = accoidmg to poweis of y, that development will be (22) F(x, y^^Aq+Aj/ + -\~A n y n i-A n + l y n+1 -f * > where the coefficients A % are power series in x, of which the first n are zero for x = 0, while A n does not vanish for x = Let C and C' be two circles of radii R and R' described in the planes of x and y respectively about the origin as center. We shall suppose that the function F(x, y) is analytic in the region defined by these two circles and also on the circles themselves , since A n is not zero for #=0, we may suppose that the radius R of the circle C is sufficiently small so that A n does not vanish in the interior of the circle C nor on the circle Let M be an upper bound for | F(x, y) \ in the preceding region and B a lower bound foi \A n \ By Cauchy's fundamental theorem we have where x and y are any two points taken in the circles C and C f ; from this we conclude that the absolute value of the coefficient A m of y m in the formula (22) is less than M/R' m } whatever may be the value of x in the circle C 234 SEVERAL VARIABLES [V, 98 We can now write (23) *(,?) where Let p be the absolute value of y , we have j BR"\S' R' 2 I BR* R' and this absolute value will be less than 1/2 if we have On the other hand, let /-c(r) be the maximum value of the absolute values of the functions A Q , A v , A n _ l for all the values of x for which the absolute value does not exceed a number r < R Since these n functions are zero for x = 0, p (r) approaches zero with r, and we can always take r so small that where p is a definite positive number The numbers r and p having been determined so as to satisfy the preceding conditions, let us re- place the circle C by the circle C r described in the as-plane with the radius r about the point x = as center, and similarly in the y-plane the circle C' by the concentric circle C' p with the radius p. If we give to x a value such that |#|=i r, and then cause the variable y to describe the circle C'p, along the entire circumference of this circle we have, from the manner in which the numbers r and p have been chosen, |P| < 1/2, | Q| < 1/2, and therefore |P + Q| < 1. If the variable y describes the circle Cp in the positive sense, the angle of 1 + P + Q returns to its initial value, whereas the angle of the factor A^f in- creases by 2 nir The equation F(x 9 y) = 0, in which | x | ^ r, therefore has n roots whose absolute values are less than p, and only n, All the other roots of the equation F(x, y) = 0, if there are any, have their absolute values greater than p Since we can replace the number p by a number as small as we wish, less than p, if we replace V, 98] IMPLICIT FUNCTIONS 235 at the same time r by a smallei numbei satisfying always the con- dition (25), we see that the equation F(x, y) = has n roots and only n which appioach zero with x If the vaiiable x lemains in the interior of the cncle C, or on its circumference, the n roots y v y 2 , , y n , whose absolute values are less than p, remain within the circle C' p These loots are not in general analytic functions of x in the cncle C,, but every symmetric integral rational function of these n roots is an analytic function of x in this cn- cle It evidently suffices to prove this foi the sum y\ -f- y% 4- 4- ?/n> where k is a positive integer Let us consider for this purpose the double integial I 7,1 II *= I d y ] v' L *S(C) *J{CA where we suppose \x\ < r If y 1 = p, the function .F (#', ?/') cannot vanish for any value of the variable x 1 within or on C r , and the only pole of the function under the integral sign m the interior of the circle C r is the point x' = x We have, then, 't/CCr} and consequently By a general theorem ( 48) this integral is equal to where ?/ 1? 2/ 2> , y n aie the n roots of the equation F(x, y) = with absolute values less than /> On the other hand, the integral / is an analytic function of x in the circle C r , for we can develop !/(#' or) in a uniformly convergent series of powers of x, and then calculate the integral term by term. The different sums 5?/? being analytic functions in the circle C r , the same thing must be true of the sum of the roots, of the sum of the products taking two at a time, and so on, and therefore the n roots y v y# , y n are also roots of an equar tion of the nth degree (26) 286 SEVERAL VARIABLES [v, 98 whose coefficients a v a 2 , , a n are analytic functions of x in the cncle C r vanishing for x = 0. The two functions F(x 9 y} and f(x 9 y) vanish f 01 the same pairs of values of the variables x, y in the interior of the circles C r and C' p We shall now show that the quotient F(x } y)/f(x, y) is an analytic function in this region Let us take definite values for these vari- ables such that |;c| O, \!/\<p, and let us consider the double integral r r . C F(x\ y 1 ) dot J = IFor a value of y 1 of absolute value p the function /(&', y') of the variable x' cannot vanish for any value of x 1 within or on the circle C r The function undei the integral sign has theiefore the single pole x' = x within C r} and the corresponding residue is Hence we have also J = but the two analytic functions F(x 3 ?/'), /(#, ?/) of the variable ?/ have the same zeros with the same degrees of multiplicity in the interior of C' p . Their quotient is therefore an analytic function of y ! in C' p , and the only pole of the function to be integrated in this circle is y' = y , hence we have On the other hand, we can replace !/(#' x) (y' y) in the inte- gral by a uniformly convergent series arranged in positive powers of x and y. Integrating term by term, we see that the integral is equal to the value of a power series pioceeding according to powers of x and y and convergent in the circles C t> C' p Hence we may write or (27) where the function H(x 9 y) is analytic in the circles C r , C' f . The coefficient A n of y in F(x, y) contains a constant term dif- ferent from zero , since a v a# - , a n are zero ^ or lffiy|^| e develop- ment of H(x y y) necessarily contains a constant tipf'i^prent from zero, and the decomposition given by the expre^(|^4l|^) throws v,99] IMPLICIT FUNCTIONS 237 into relief the fact that the roots of F(x, y)= which approach zero with x aie obtained by putting the first factor equal to zero The preceding important theorem is due to Weierstrass.* It generalizes, at least as far as that is possible for a function of several variables, the decomposition into factors of functions of a single variable. 99. Critical points. In order to study the n roots of the equation F(x, y) = which become infinitely small with x, we are thus led to study the roots of an equation of the form foi values of x near zero, where a^ a 2 , -, a n are analytic functions that vanish for x = When n is greater than unity (the only case which concerns us), the point x = is in geneial a critical point. Let us eliminate y between the two equations / = and df/dy = , the resultant A (x) is a polynomial in the coefficients a v a^ , a n , and therefore an analytic function in the neighborhood of the ongm. This resultant t is zeio for x = 0, and, since the zeros of an analytic function form a system of isolated points, we may suppose that we have taken the radius r of the circle C r so small that in the interior of C r the equation A (x) = has no other loot than x = 0. Tor every point X Q taken in that circle othei than the origin, the equation f( x o) 2/) = will have n distinct roots According to the case already studied (I, 194, 2d ed ; 188, 1st ed ), the n roots of the equation (28) will be analytic functions of x in the neighborhood of the point # Hence there cannot be any other critical point than the origin in the intenor of the encle C r . Let y t , y^ -, y n be the n roots of the equation /(cc , ?/) = 0. Let us cause the variable x to describe a loop around the point x = 0, starting from the point X Q , along the whole loop the n roots of the equation f(x, y) = are distinct and vary in a continuous mannei. If we start from the point X Q with the root y v for example, and fol- low the continuous variation of that root along the whole loop, we return to the point of departure with a final value equal to one of the roots of the equation /(cc , y) = If that final value is y v the root * Abhandlungen aus der Functionerilehre von K Wezerstrass (Berlin, 1860) The proposition can also be demonstrated by making use only of the properties of power series and the existence theorem for implicit functions (Bulletin de la Soat6 mathematique, Vol XXXVI, 1908, pp 209-215) t We disregard the case where the resultant is identically zero In this case / (x, y) would be divisible by a factor [fi(x, y)]*, where k > 1, fi(z 9 y) being of the same form as /(a, y) 238 SEVERAL VARIABLES [V,99 consideied is single-valued in the neighborhood of the origin If that final value is different from y^ let us suppose that it is equal to y 2 . A new loop descubed in the same sense will lead from the root y 2 to one of the roots y v y^ - , y n The final value cannot be y^ since the reverse path must lead from y 2 to y^ That final value must, then, be one of the loots y^ y^- -, y n If it is y^ we see that the two roots y l and y 2 are permuted when the vanable describes a loop around the origin If that final value is not y^ it is one of the remaining (n 2) roots , let y s be that root A new loop descubed in the same sense will lead from the root y g to one of the loots y l9 y 2 , y 3 , y 4 , - , y n It cannot be y g , for the same reason as before , neither is it y 2 , since the reverse path leads from y 2 to y^ Hence that final value is either y l or one of the remaining (n 3) roots y^ y 5 , , y n If it is y 1? the three loots y v y 2 , y z permute themselves cyclically when the variable x describes a loop around the origin, If the final value is different from y^ we shall continue to cause the vanable to turn around the origin, and at the end of a finite number of operations we shall necessarily come back to a root already obtained, which will be the root y l Suppose, for exam- ple, that this happens after p operations , the p roots obtained, 2^i? y& *> %>? permute themselves cyclically when the variable x describes a loop around the origin We say that they form a cyclic system ofp roots If p = n, the n roots form a single cyclic system If p is less than n, we shall repeat the reasoning, starting with one of the remaining n p roots and so on. It is clear that if we con- tinue in this way we shall end by exhausting all the roots, and we can state the following proposition. The n roots of the equation F(x, y) = 0, which are zero for x = 0, form one or several cyclic systems in the neighborhood of the origin To render the statement perfectly general, it is sufficient to agree that a cyclic system can be composed of a single root ; that root is then a single-valued function in the neighborhood of the origin The roots of the same cyclic system can be represented by a unique development. Let y v y z , -, y p be the p roots of a cyclic system , let us put x = x*. Each of these roots becomes an analytic function of x' for all values of x ! other than y} = , on the other hand, when x 1 describes a loop around x r = 0, the point x describes p succes- sive loops m the same sense around the origin. Each of the roots 2/i? y 2 > '> VP returns then to its initial value , they are single-valued functions in the neighborhood of the origin Since these roots ap- proach zero when x 1 approaches zero, the origin x' = cannot be V, 99] IMPLICIT FUNCTIONS 239 otliei than an ordinary point, and one of these roots is represented by a development of the form (29) 2/ = X + *X 2 + '* +<v"-f..., or, replacing x* by x l/p , i / i\ 2 / iV (30) y = ^a* + a^ap; + + a m \x) + . We may now say that the development (30) represents all the roots of the same cyclic system, provided that we give to x lfp all of its p determinations For, let us suppose that, taking for the radical V& one of its determinations, we have the development of the loot y r If the variable x describes a loop around the origin in the positive sense, y^ changes into y^ and x l/p is multiplied by e 2lrt/ * It will be seen, similarly, that we shall obtain y q by replacing x l/p by x l/p e zq1ri/p in the equality (30) This unique development for the system shows up clearly the cyclic permutation of the^ roots It would now lemain to show how we could separate the n roots of the equation F(x, y) = into cyclic systems and calculate the coefficients a t of the develop- ments (30) We have already considered the case where the point x = y = is a double point (I, 199, 2d ed ) We shall now treat another particular case If f 01 x = y = the derivative dF/dx is not zero, the develop- ment of F(x, y) contains a term of the first degree in x, and we have (31) F(x,y) = 4x+Bf + , (AB*0) where the terms not written are divisible by one of the factors cc 2 , vy, y n+l Let us consider y for a moment as the independent vanable; the equation F(x, y) = has a single root approaching zero with y, and that root is analytic in the neighborhood of the origin The development which we have already seen how to calculate (I, 35, 193, 2d ed , 20, 187, 1st ed.) runs as follows (32) s = 0"(*o + iy+ ) (*<>) Extracting the wth root of the two sides, we find (33) af For y = the auxiliary equation u n == a Q + a^j + has n dis- tinct roots, each of which is developable in a power series according to powers of y. Since these n roots are deducible from one of them by multiplying it by the successive powers of &* m/n , we can take for ~\/a, Q + a~y + in the equality (33) any one of these roots, subject to the condition of assigning successively to x l/n its n determinations 240 SEVERAL VARIABLES [V, 99 We can therefore write the equation (33) in the form a = i 1 y + a i y a + , ft *> 0) and from this ve deiive, conversely, a development of y in poweis (34) ^ = ^ + This development, if we give successively to # Vn its n values, represents the n roots which appioach zero with x These n roots form, then, a single cyclic system Foi a study of the general case we lefei the reader to treatises devoted to the theory of algebraic functions * 100. Algebraic functions Up to the present time the implicit func- tions most carefully studied aie the algebraic functions, defined by an equation F(x : y) = 0, in which the left-hand side is an irreducible polynomial m x and y A polynomial is said to be irreducible when it is not possible to find two other polynomials of lower degiee, F^x, y) and F 2 (#, y), such that we have identically F(x 9 y) = F&, y) X Fx, y). If the polynomial F(x, y) were equal to a product of that kind, it is clear that the equation F(x, y) = could be replaced by two distinct equations Ffo, y) = 0, F 2 (x, y) = Let, then, (35) F(x,y) = 4>,(xW+4> l (z)p-' i + + *,-i(B)y+* i (aO=0 be the proposed equation of degree n in ?/, where < , $ v , <f> n are polynomials in % Eliminating ?/ between the two relations F = 0, dF/dy = 0, we obtain a polynomial A(OJ) for the resultant, which can- not be identically zero, since F(x, y) is supposed to be irreducible. Let us rnaik in the plane the points a v a,, , (%, which represent the roots of the equation A(x)= 0, and the points p v yS 2 , - , p h , which represent the roots of < (&)= Some of the points <r z may also be among the roots of < (&)= For a point a different from the points a l3 fa the equation F(a,y*)= has n distinct and finite roots, # 1? 5 2 , * - -, b n In the neighborhood of the point a the equation (35) has therefore n analytic roots which approach b v & 2 , , b n respectively when x approaches a. Let a % be a root of the equation * See also tlie noted memoir of Puaseux on algebraic functions (Journal de MatM- mattques, Vol XV, 1850) V,100] ALGEBRAIC FUNCTIONS 241 A (#) = The equation F(a iy y) = has a certain numbei of equal roots , let us suppose, for example, that it has p roots equal to I The p roots which appioach Z> when x appi caches tr, group themselves into a ceitam numbei of cyclic systems, and the roots of the same cyclic system are represented by a development 111 series arranged accoiding to fiactional powers of x # t If the value a t does not cause <j5> (&) to vanish, all the roots of the equation (35) in the neigh- boihood of the point a l group themselves into a certain number of cyclic systems, some of which may contain only one root For a point fa which makes < (#) zero, some of the loots of the equation (35) become infinite , in order to study these roots, we put y == !//, and we are led to study the roots of the equation which become zeio for x = fa These roots group themselves again into a certain number of cyclic systems, the loots of the same system being repiesented by a development in senes of the form (36) y' = o(a - &) * + a m+l (x - ft)^ + > K * 0) The coiresponding roots of the equation in y will be given by the development (37) y = (x - fl)~*[rti. + *(* - ft) >+ which can be arranged in increasing powers of (r J3,) l/p , but there will be at first a finit e number of terms with negative exponents. To study the values of y f 01 the infinite values of x, we put x = l/# f , and we are led to study the roots of an equation of the same form in the neighborhood of the origin To sum up, in the neighborhood of any point x = a the n roots of the equation (35) are represented by a certain number of senes arranged according to increasing powers of x a or of (x a) l/p , containing perhaps a finite numbei of terms with negative exponents, and this statement applies also to infinite values of x by replacing x oc by 1/cc. It is to be observed that the fractional powers or the negative ex- ponents present themselves only for the exceptional points The only singular points of the roots of the equation aie therefore the critical points around which some of these roots permute themselves cyclically, and the poles where some of these roots become infinite ; moreover, a point may be at the same time a pole and a critical point These two kinds of singular points are often called algebraic singular po^nts. 242 SEVERAL VARIABLES [V, 100 We have so far studied the roots of the proposed equation only in the neighborhood of a fixed point Suppose now that we ]0in two points x = a, x = b, for which the equation (35) has n distinct and finite roots, by a path AB not passing through any smgulai point of the equation Let y^ be a root of the equation F(a, y) = , the root y =/(x), which reduces to y l for x = a, is represented in the neigh- borhood of the point a, by a power-series development P (x a) We can propose to ourselves the problem of finding its analytic ex- tension by causing the variable to describe the path AB This is a particular case of the general problem, and we know in advance that we shall arrive at the point B with a final value which will be a root of the equation F(b, y) = ( 86) We shall surely arrive at the point 1} at the end of a finite number of operations , in fact, the radii of the circles of convergence of the series representing the different roots of the equation F(x, y) = 0, having their centers at different points of the path AB } have a lower limit* 8 >0, since this path does not contain any critical points ; and it is clear that we could always take the radii of the different circles which we use for the analytic extension at least equal to 8 Among all the paths joining the points A and B we can always find one leading from the root y^ to any given one of the roots of the equation F(l, y) = as the final value The proof of this can be made to depend on the following proposition If an analytic func- tion & of the variable x has only p distinct values for each value ofx, and if it has in the whole plane (including the point at infinity) only algebraic singular points, the p determinations of z are roots of an equation of degree p whose coefficients are rational functions of x Let s 1? z 2 , - , z p be the p determinations of z , when the variable x describes a closed curve, these p values z v z^ , # p can only change into each other. The symmetric function u k = z\ 4- 4, + + > where & is a positive integer, is therefore single-valued Moreover, that function can have only polar singularities, for in the neigh- borhood of any point in the finite plane x = a the developments ot z v s 2 , - -, p have only a finite number of terms with negative exponents. The same thing is therefore true of the development of %. Also, the function % being single-valued, its development cannot con- tain fractional powers The point a is therefore a pole or an ordinary point for %., and similarly for the point at infinity. The function u k * To prove this rigorously it suffices to make use of a form of reasoning analogous to that of S 84 V, 101] ALGEBRAIC FUNCTIONS 243 is therefore a rational function of x, whatever may be the integer 70, consequently the same thing is true of the simple symmetric functions, such as S,, S^,*?*, > which proves the theoiem stated Having shown this, let us now suppose that 111 going from the point a to any other point x of the plane by all possible paths we can obtain as final values only p of the roots of the equation F(x,y)=Q, CP<) These p roots can evidently only be permuted among themselves when the variable x describes a closed boundaiy, and they possess all the properties of the p branches 19 & 2) , & p of the analytic function which we have just studied. We conclude from this that T/ I? y^ , y p would be roots of an equation of degree p, F^x, y) = 0, with rational coefficients The equation F(x f y) would have, then, all the roots of the equation Ffo, y) = 0, whatever x may be, and the polynomial F(x, y) would not be irreducible, contrary to hypothesis If we place no restriction upon the path followed by the variable x, the n roots of the equation (35) must then be regarded as the distinct branches of a single analytic function, as we have already remarked in the case of some simple examples (6). Let us suppose that from each of the critical points we make an infinite cut in the plane in such a way that these cuts do not cross each other. If the path followed by x is required not to cross any of these cuts, the n roots are single-valued functions in the whole plane, for two paths having the same extremities will be transform- able one into the other by a continuous deformation without passing over any critical point ( 85) In order to follow the variation of a root along any path, we need only know the law of the permutation of these roots when the variable describes a loop around each of the critical points Note The study of algebraic functions is made relatively easy by the fact that we can determine a priori by algebraic computation the singular points of these functions This is no longer true in general of implicit functions that are not algebraic, which may have transcendental singular points As an example, the implicit function y (x) , defined by the equation & # 1 = 0, has no algebraic critical point, but it has the transcendental singular point x = 1. 101 . Abelian integrals. Every integral I=>fR (a?, y) dx, where E (as, y) is a rational function of x and y, and where y is an algebraic func- tion defined by the equation F(x, y) = 0, is called an Abelian integral attached to that c^trve. To complete the determination of that inte- gral, it is necessary to assign a lower limit X Q and the corresponding 244 SEVERAL VARIABLES [V, 101 value ?/ chosen among the roots of the equation F(x Q , */)= We shall now state some of the most important geneial properties of such integials When we go from the point x to any point x by all the possible paths, all the values of the mtegial I are included in one of the formulae (38) JT:=7 Ji + m 1 ai 1 + m a tt a + + > r <D r , (& = 1>2> - ,72,) where 7 13 7 2 , , I n are the values of the integral which correspond to certain definite paths, m v m 2 , - , m r are arbitrary integers, and o> 1? a>>, , o>, are penods These periods are of two kinds , one kind results fiom loops described about the poles of the function It (a, y) , these are the polar periods The others come from closed paths surrounding several critical points, called cycles, these are called cyclic periods. The numbei of the distinct cyclic periods depends only on the algebraic relation considered, F(r, y) , it is equal to 2p, where p denotes the deficiency of the cuive ( 82) On the other hand, there may be any number of polar periods From the point of view of the singularities three classes of Abelian integrals aie distinguished Those which remain finite in the neighborhood of every value of x are called the first kind-, if their absolute value becomes infinite, it can only happen thiough the addition of an infinite number of periods The integrals of the second kind are those which have a single pole, and the integrals of the third Jcmd have two logarithmic singular points Every Abelian integral is a sum of integrals of the three kinds, and the number of distinct integials of the first kind is equal to the deficiency The study of these integrals is made very easy by the aid of plane surfaces composed of several sheets, called Riemann surfaces We shall not have occasion to consider them here We shall only give, on account of its thoroughly elementary character, the demonstrar tion of a fundamental theorem, discovered by Abel 102. Abel's theorem. In order to state the results more easily, let us consider the plane curve C represented by the equation F(x, y) = 0, and let $ (x, y) be the equation of another plane algebraic curve C r These two curves have N points in common, (x v y^), (# 2 , y^, , ( x m VN)) the number N being equal to the product of the degrees of the two curves Let R(x, y) be a rational function, and let us consider the following sum (39) jW JZ(x,y)fc, V,102] ALGEBRAIC FUNCTIONS 245 where /&*? R(x,y}dx /&*?) I / (*o yd denotes the Abelian integral taken from the fixed point X Q to a point x along a path which leads y from the initial value y to the final value y^ the initial value ?/ of y being the same for all these integrals It is clear that the sum / is determined except f 01 a period, since this is the case with each of the integrals Suppose, now, that some of the coefficients, a v a^ , a l} of the polynomial <(#, y) are variable When these coefficients vary continuously, the points # t themselves vary continuously, and if none of these points pass through a point of discontinuity of the integral fR (x, y) dx } the sum J itself varies continuously, provided that we follow the continuous variation of each of the integrals contained in it along the entire path described by the corresponding upper limit. The sum / is therefore a function of the parameters a v a 2 , , a ly whose analytic form we shall now investigate Let us denote in general by 87 the total differential of any func- tion V with respect to the variables & 1? & 2 , , a L : By the expression (39) we have From the two relations F(x l7 y^) = 0, $(x t , 2/ t ) = we derive and consequently 8# t = V(x % , 2/ t )8$ t , where ^(x^y^) is a rational function of x l} y^ a v a^ , a L , and where <E> t is put for We have, then, ^ N The coefficient of 8^ on the right is a rational symmetric function of the coordinates of the N points (x l} y^) common to the two curves C, C" The theory of elimination proves that this function is a rational function of the coefficients of the two polynomials F(x, y) and $(#, y) y and consequently a rational function of a v a z , , a k Evidently the same thing is true of the coefficients of 8& 2 , , 8^, 246 SEVERAL VARIABLES [V,102 and I will be obtained by the integration of a total differential / = f where TT I? IT,, , IT L aie lational functions of a v c& 2 , , % Now the integration cannot introduce any other tianscendentals than logarithms. The sum I is therefore equal to a, rational function of the coefficients a lt a v , a l9 plus a sum of logarithms of rational functions of the same coefficients, each of these logarithms being 'multiplied bi/ a constant factor This is the statement of Abel's theorem in its most general form In geometiic language we can also say that the sum of the values of any Abelian integral, taken from a common origin to the N points of intersection of the given curve with a variable curve of degree m, $(x, y)= 0, is equal to a rational function of the coefficients of *(x, y),plu* <*> sum of a finite number of logarithms of rational functions of the same coefficients, each logarithm being multiplied by a constant factor The second statement appears at first sight the more striking, but in applications we must always keep in mind the analytic state- ment in the evaluation of the continuous variation of the sum / which corresponds to a continuous vanation of the parameters a lt a a , , a L . The theorem has a precise meaning only if we take into account the paths described by the N points x l9 a? a , , X N on the plane of the variable x. The statement becomes of a remarkable simplicity when the integral is of the first kind In fact, if TT I? 7T 2 , , TT^ were not identically zero, it would be possible to find a system of values a^ = a{, , a k = a{ for which I would become infinite. Let (arj, yj), . , (tfy, 1/tf) be the points of intersection of the curve C with the curve C' which correspond to the values a[ , , a' K of the parameters The integral ^ I /( would become infinite when the upper limit approaches one of the points (o, 2/0* which is impossible if the integral is of the first kind Therefore we have S/ = 0, and, when a v a^ , % vary continuously, / remains constant , Abel's theorem can then be stated as follows . Given a fixed curve C and a variable curve C 1 of degree m, the sum of the increments of an Abelian integral of the first kind attached to the curve C along the eontmuous curves described by the points of intersection of C with C 1 is equal to zero. V,loa] ALGEBRAIC FUtfCTIOSTS 247 Note We suppose that the degree of the curve C* remains con- stant and equal to m If for certain particular values of the coeffi- cients a v & 2 , , a k that degree weie lowered, some of the points of intei sections of C with C' should be regarded as thrown off to infinity, and it would be necessary to take account of this in the application of the theorem We mention also the almost evident fact that if some of the points of intersection of C with C" are fixed, it is unnecessaiy to include the corresponding integrals in the sum L 103 Application to hyperelliptic integrals The applications of AbePs theorem to Analysis and to Geometry are extremely numer- ous and important We shall calculate SI explicitly in the case of hypeielliptic integrals. Let us consider the algebraic relations (40) ^ = (aO= where the polynomial R(x) is prime to its derivative We shall suppose that A Q may be zero, but that A Q and A l may not be zero at the same time, so that R (x) is of degree 2p + 1 or of degree 2^ + 2 Let Q(x) be any polynomial of degree q We shall take for the initial value X Q a value of x which does not make R (x) vanish, and f 01 y a root of the equation y 1 = R (a? ). We shall put where the integral is taken along a path going from x to x, and where y denotes the final value of the radical V,R(a;) when we start from x with the value y Iw order to study the system of points of intersection of the curve C represented by the equation (40) with another algebraic curve C', we may evidently replace in the equation of the latter curve an even power of y, such as f r > by [-K()] r , and an odd power y* r+i by y[^()T These substitutions having been made, the equation obtained will now contain y only to the first degree, and we may suppose the equation of the curve C' of the form (41) where /(a?) and < (x) are two polynomials prune to each other, of degrees X and /* respectively, some of the coefficients of which we shall suppose to be variable The abscissas of the points of intersec- tion of the two curves C and C' are roots of the equation (42) j( 248 SEVERAL VARIABLES [V, 103 of degree N. For special systems of values of the variable coefficients m tlie two polynomials f(x) and < (x) the degiee of the equation may turn out to be less than A 7 , some of the points of intersection are then thiown off to infinity, but the corresponding integrals must be included in the sum which we aie about to study To each root x l of the equation (42) corresponds a completely determined value of y given by y=/(i)/^(^i) ^et us now consider the sum We have for the final value of the radical at the point x t must be equal to ?/ 1? that is, to f(x t )/<j> (a 1 ,) On the other hand, from the equation ^(aj t )= we derive </,'(*,) te. + 2 * (as,) < (*,) 8*. - 2/fo) 8/, = 0, and therefore or, making use of the equation (42), Let us calculate, for example, the coefficient of Sa k in 87, where a fc is the coefficient, supposed variable, of x 1 in the polynomial f(x) The term 8% does not appear in 8$,, and it is multiplied by x% in 8/ t The desired coefficient of 8a k is therefore equal to a * _ where ?r(a;) = Q (a;) <jf> (#) aj x . The preceding sum must be extended to all the roots of the equation i/r(x) = , it is a rational and symmetric function of these roots, and therefore a rational function of the coeffi- cients of the two polynomials f(x) and < (x). The calculation of this sum can be facilitated by noticing that STT^)/^'^) is equal to the sum of the residues of the rational function 7r(x)/iff(x) relative to the N poles in the finite plane x v j 2 , ,X N By a general theo- rem that sum is also equal to the residue at the point at infinity with its sign changed ( 52) It will be possible, then, to obtain the coefficient of 8a L by a simple division. V, 103] ALGEBRAIC FUNCTIONS 249 It is easy to prove that this coefficient is zero if the integral v(cc, y) is of the first kind We have by supposition q ^ p 1 , the degree of TT(CC) is q -h p + k, and we have Let us find the degree of ifr(x). If there is no cancellation between the terms of highest degree in R(x) <^(x) and m/ 3 (o;), we have 2\ whence and, a fortiori, If there were a cancellation between these two terms, we should have A, =: but since the term a k x^^ k has no term with which to cancel out, we should have \ + & ^ N, from which the same inequality as before lesults It follows that we always have The residue of the rational function 7r(x)/ij/(x) with respect to the point at infinity is therefore zero, for the development will begin with a term in 1/x 2 or of higher degree. It will be seen similarly that the coefficient of Bb h in 87, b h being one of the variable coefficients of the polynomial <(#), is zero if the polynomial Q(x) is of degree p 1 or of lower degree This result is completely in accord with the general theorem. Let us take, for example, < (x) = 1 3 and let us put a p x* wheie a , a l9 , a p are p + 1 variable coefficients The two curves cut each other in 2p + 1 variable points, and the sum of the values of the integral v (z, y), taken from an initial point to these 2 p + 1 points of intersection, is an algebraic-logarithmw function of the coefficients a , a l9 , a p Now we can dispose of these p -f 1 coeffi- cients in such a way that p -f 1 of the points of iDtersection are any previously assigned points of the curve ^=^R(x)^ and the coordi- nates of the p remaining points will be algebraic functions of the coordinates of the p + 1 given points. 250 SEVERAL VARIABLES [V, 103 The sum of the^> + 1 integrals taken from a common initial point to p 4- 1 arbitrary points, is therefore equal to the sum of p integrals whose limits are algebraic functions of the coordinates plus certain algebiaic-logarithmic expressions It is clear that by successive reductions the proposition can be extended to the sum of m integrals, where m, is any integer gi eater thanj? In particular, the sum of any number of integrals of the first kind can be i educed to the sum of only p integrals. This propeity, which applies to the most general Abelian integrals of the first kind, constitutes the addition theorem for these integrals. In the case of elliptic integrals of the first kind, Abel's theorem leads pre- cisely to the addition formula for the function p(u) Let us consider a cubic in the normal form and let Mjfa, y^ 3f 2 (x 2 , y a ), ^ S (x 3 , 2/ 8 ) be the points of intersection of that cubic with a straight line D By the general theorem the sum is equal to a period, for the three points M v Jf s , 3f 3 are carried off to infinity when the straight line D goes off itself to infinity Now if we employ the parametric representation x =p(w), y = P'(M) for the cubic, the parameter u is precisely equal to the integral I - ._ and the preceding formula says that the sum of the arguments M I? u 2 , w 8 , which correspond to the three points M v Jf 2 , Jkf s , is equal to a period We have seen above how that relation is equivalent to the addition formula for the function p(u)(80). 104. Extension of Lagrange's formula The general theorem on the implicit functions defined by a simultaneous system of equations (I, 194, 2d ed , 188, 1st ed) extends also to complex variables, provided that we retain the other hypotheses of the theorem. Let us consider, for example, the two simultaneous equations (44) P(a,y) = fc-a-a/(x,y) = 0, Q(ai, y) = y-6- p<l>(x, y) = 0, where x and y are complex variables, and where /(<c, y) and <f> (x, y) are ana- lytic functions of these two variables in the neighborhood of the system of V,104] ALGEBRAIC FUNCTIONS 251 values x = a, y = b For a = 0, /3 = these equations (44) have the system of solutions x = a, y = 6, and the deteiminant D(P, Q)/D(x, y) leduces to unity Theiefoie, by the general theorem, the system of equations (44) has one and only one system of roots appi caching a and 6 lespectively when a and /3 approach zero, and these loots are analytic functions oi a and p Laplace was the first to extend Lagrange's formula ( 51) to this system ot equations Let us suppose for defimteness that with the points a and & as centers we describe two circles C and <7' in the planes of the variables x and y respectively, with ladn r and r 7 so small that the two functions /(x, y) and <f> (x, y) shall be analytic when the vanables x and y remain within 01 on the boundaries of these two ciicles (7, C' Let M and M' be the maximum values of |/(x, y) \ and of |0(x, y)|, lespectively, in this region We shall suppose fuither that the constants a and /5 satisfy the conditions M \ a \ < r, If' | /3 1 < r 7 Let us now give to x any value within 01 on the boundary of the ciicle (7, the equation Q (x, y) = is satisfied by a single value of y in the interior of the circle <7', for the angle of y & (x, y) increases by 2 TT when y describes the circle C' in the positive sense ( 49) That root is an analytic function y l = ^ (x) of x in the circle (7 If we replace y in P (x, ?/) by that root y t , the resulting equation x a ar/(x, y a ) = has one and only one root in the inte- 1101 of (7, for the reason given a moment ago Let x = be that root, and let t\ be the corresponding value of y, t\ = ^ () The object of the generalized Lagrange formula is to develop in powers of a, and /3 eveiy function Ffa 17) which is analytic m the region just defined For this purpose let us consider the double integral MK\ r C A* C fffe y)&y (QtO) JL s= I UfX I Since x is a point on the circumference of (7, P(x, y) cannot vanish for any value of y within C', for the angle of x a a/(x, y) returns necessarily to its initial value when y describes <7 X , x being a fixed point of C The only pole of the function under the integral sign, considered as a function of the single variable y, is, then, the point yy v given by the root of the equation Q (x, y) = 0, which corresponds to the value of x on the boundary C, and we have, after a first integration, C F(x^y)dy >. ff(x, yi) Jcc")P(a, v) Qfa y) " ^ p/ v x /a? v ^ i; \dy The right-hand side, if we suppose y^ replaced by the analytic function $ (x) defined above, has in turn a single pole of the first order in the interior of C, the point x = ft to which corresponds the value y l = % and the corresponding residue is easily shown to be The double integral I has therefore for its value I=-4* rj ( p f yi '* 252 SEVEEAL VARIABLES [V, 104 On the other hand, we can develop l/PQ m a uniformly conveigent series b-p<p) <*-i (x - which gives us I =SJ r mn a/3, where r = r fa r -Ffe y) E/te y)] m E* to y)?^ mn J<C) J7) (aj-a)+i(tf-&)+ 1 This integral has already heen calculated ( 94), and we have found that it is 47T 2 d Equating the two values of I, we obtain the desired result, which presents an evident analogy with the formula (50) of 51 "We could also obtain a second result analogous to (51), of 51, by putting but the coefficients in this case are not so simple as in the case of one variable EXERCISES 1, Every algebraic curve C n of degree n and of deficiency jp can be earned over by a birational transformation into a curve of degree p + 2 (Proceed as in 82, cutting the given curve by a net of curves <7 n _ 2 , passing through n (n 1)/2 3 points of (7 n , among which are the (n 1) (n 2)/2 p double points, and put the equation of the net being ^ x (aj, y) + X0 2 (a>, y) + ^0 8 (aj, y) = 0.) 2 Deduce from the preceding exercise that the coordinates of a point of a curve of deficiency 2 can be expiessed as rational functions of a parameter t and of the square root of a polynomial R(t) of the fifth or of the sixth degiee, prime to its derivative (The reader may begin by showing that the curve corresponds point by point to a curve of the fourth degree having a double point ) 3* Let y = a^x + <x z x 2 + - be the development in power series of an alge- braic function, a root of an equation F(x, y) = 0, where F(x, y) is a polynomial with integral coefficients and where the point with coordinates x = 0, y = is a simple point of the curve represented by Ffay) = Q All the coefficients e^, <ar a are fractions, and it suffices to change x to Jfo, K being a suitably chosen integer, in order that all these coefficients become integers, [EISENSTEIN ] (It will be noticed that a transformation of the form x = Wx', y ky' suffices to make the coefficient of y" on the left-hand side of the new relation equal to one, all the other coefficients being integers.) INDEX [Titles m italic are proper names, immbeis in italic are page numbers , and num- bers in roman type are paragraph numbers ] Abel : 19, f tn , 170, 76 , 180, 78 , 198, 82 , 244, 101 Abelian mtegials : see Integrals Abel's theorem : 244, 102 Addition formulae : 27, 12; for elliptic functions : 166, 74 , $50, 103 Adjoint curves : 191, 82 Affixe : 4, f tn Algebraic equations : see Equations Algebraic functions : see Functions Algebraic singular points : see Singular points Analytic extension : 196, 83 , 199, 84 ; functions of two variables : 31, 97 Analytic functions: 7, 3 , 11, 4; ana- lytic extension: 196, 83, 199, 84 J derivative of : 9, 3, 48, 9, 77, 33; elements of : 198, 83 ; new definition of : 199, 84 ; series of : 86, 39 ; zeros of : 88, 40 , see also Cauchy's theo- rems, Functions, Integral functions, Single-valued analytic functions Analytic functions of several varia- bles: 218, 91; analytic extension of: 281, 97; Cauchy's theorems: 225, 94, 227, 95; Lagrange's for- mula : 250, 104 ; Taylor's formula : 222, 92 , 226, 94 ; singularities of : 232, 97 Anchor ring : 54, ex 3 Appell : 84, 38 , 217, ex. 3 Associated circles of convergence : 220, 92 Associated integral functions: 218, ex 7 Bertrand : 58, ex 22 Bicircular quartics : 198, ex. Binomial formula : 40, 18 Birational transformations : 192, 82 , 252, ex 1 Blumenthal : 188, f tn. Borel: 180, ftn , 138, ftn , .?, ex. 3 Bouquet . see Btiot and Bouquet Branch point: see Critical points Branches of a function : 15, 6 , 29, 13 Bnot and Bouquet : 126, ex 27 , 195, ex 11 Burman: 126, ex 26 Burman's series : 126, ex 26 7, 2 , 0, ftn , j?0, 3 ; 60, 25 , 7jf, ftn , 74, 32 , 7*, 34 ; 82, ftn , 106, 51 , ^4 5 53 , 127, 57 , .Z30, 63 , 200, ftn , 216, 90 , &8, 93 , 886, 94 , jft*7, 95 , 838, 98 Cauchy-Laurent series : 81, 35 Cauchy's integral* 75, 33; funda- mental formula: 76, 33, 7, 95; fundamental theorem : 283, 98 , in- tegral theorems: 75, 33; method, Mittag-Leffler's theorem: 189, 63; theorem: 66, 28, 7-?, ftn , 74, 32, 75, 33 , 78, 34 , 816, 90; theorem for double integrals : 888, 93 , 886, 94 Cauchy-Taylor series : 79, 35 Change of variables, in integrals : 62, 26 Circle of convergence : 18, 8 , 808, 84, #00, 87, 818, 88; associated circles of convergence: 880, 92; singular points on : 808, 84 and ftn Circular transformation : 45, 19 , 57, ex 13 Class of an integral function : 182, 58 Clebschi 186, ftn. Complex quantity : 8, 1 Complex variable: 6, 2; analytic func- tion of a : 9, 3 ; function of a : 6, 2 253 254 INDEX Conf ormal maps : see Maps Confoimal repiesentation : 4%, 19 , 45, 20 , 48, 20 , 52, 23 , see also Projec- tion and Transformations Conf ormal transformations: seeTians- formations Conjugate imaginanes: 4, I Conjugate isothermal systems : 54, 24 Connected region . 11, 4 Continuity, of functions : 6, 2; of power series 7, 2 , 56, ex 7 Continuous functions : see Functions Convergence, circle of: see Circle of convex gence Convergence, unifoim: of infinite products: 82, 10 , 129, 57; of inte- grals : 289,M ; of series . 7, 2 , 88, 39 Cousin : 232, f tn Critical points : 15, 6 , 0, 13 , 837, 99; logarithmic: 88, 14, 113, 53 Cubics : see Curves Curves, adjoint: 191, 82; bicircular quartics: 193, ex ; conjugate iso- thermal systems : 54, 24 ; deficiency of : 172, 77 f 191, 82 , 25%, exs 1 and 2 , 244, 101 ; of deficiency one : 178, 77 ; double points : 184, 80, 191, 82 ; loxodromic: #, ex 1, parametnc representation of curves of defi- ciency one : 187, 81 , 191, 82 , 19S, ex ; parametric representation of plane cubics . 180, 78 , 184, 80 , 187, 81; points of inflection: 186, 80; quartics : 187, 81 ; unicursal : 191, 82, see also Abel's theorem and Rhumb lines Cuts : 08, 87 Cycles: 244, 101 Cyclic periods : 244, 101 Cyclic system of roots : 238, 99 D'Alemberti 104, Note D' Alembert's theorem : 104, Note Darboux : 64, 27 Darboux's formula, law of the mean : 64,27 Deficiency : see Curves, deficiency of Definite integrals: 60, 25, 72, 31, 97, 46; differentiation of: 77, 33; 27, 95; evaluation of: 96, 45; FresnePs : 100, 46 ; F function : 100, 47 , 249, 96 ; law of the mean : 64, 27; penods of. 118, 53, ^4, Note, see also Integials De Moivre : 6, 1 De Moivre's formula, 6, 1 Denvative, of analytic functions: 9, 3 , 42, 19 , 77, 33 ; of integrals . 77, 33, 287, 95, of power senes. 19, 8; of senes of analytic functions: 8S,B9 Dominant function : 56, ex 7 , 81, QK 00v 04. oo , &&/, y* Dominant senes . &Z, 9 , 157, 69 Double integrals : 888, 93 ; Cauchy's theorems : 882, 93 , 285, 94 Double points : 184, 80 , ^PJf, 82 Double series : 81, 9 ; circles of con- vergence: 830, 92; Tayloi's for- mula : 882, 92 , S86, 94 Doubly periodic functions' 145, 65, 149, 67 , see also Elliptic functions Eisenstem : 252, ex 3 Elements of analytic functions : 198, 83 Elliptic functions: 145, 65, 150, 08; addition formulae: 1#, 74; alge- braic relation between elliptic func- tions with the same periods: 15S 68 ; application to cubics : 180, 78 , 184, 80; application to curves of deficiency one : 187, 81 , 191, 82 ; application to quartics: 187, 81; even and odd : 154, 68 ; expansions for : 154, 69 , general expression for: 168, 73; Hermite's formula: 165, 73 , 168, 75 , 195, ex 9 ; integration of : 168, 75 ; invariants of : 158, 70 , 172, 77 , 182, 79 ; order of : 150, 68 ; p(w) : 154, 69; p(u) defined by in- variants : 182, 79 ; periods of : 158, 68 ; 172, 77 , 184, 79 ; poles of : 150, 68 , 154, 68 ; relation between p(u) and p' (it) : 158, 70 ; residues of : 151, 68; <r(u) : Jft*, 72; 0,(u) : 170, 76; r(u): 150, 71; zeros 6f: jtf*, 68 , 154, 68 , jtf0, 70 INDEX 255 Elliptic integrals, of the fiist kind: 120, 56 , 174, 78 , 250, 103 ; the in- verse function : 174, 78 ; periods of : 120, 56 Elliptic transformation : $7, ex 15 Equations: 283, 98; algebraic: 240, 100 , cyclic system of roots of : 238, 99, 241, 100; D'Alembert's theo- rem: 104, Note; Keplei's: 109, ex , 126, ex 27 ; Laplace's : 10, 3 , 54, 24 , 55, Note ; theory of equa- tions : 10$, 49 , see also Implicit functions, Lagrange's formula, and Weierstrass's theorem Essentially singular point : 91, 42 ; at infinity : 1 10, 52 ; isolated : 91, 42 ; see also Laurent's senes Eul& : 27, 12 , 58, exs 20 and 22 , 96, 45, 124, ex 14, 143, ftn., 830, 96 Euler's constant: 230, 96; formula: 58, ex 22, 96, 45, 124, ex 14, formulae : 27, 12 Evaluation of definite integrals : see Definite integrals Even functions : 153, Notes Expansions in infinite products: 194, exs 2 and 3 ; of cos z : 194, ex 3; of T(z). 230, 96; of <r(u): 162, 72 ; of sin cc :' 143, 64 , see also Functions, primary, and Infinite products Expansions in series : of ctn x : 143, 64 ; of elliptic functions : 154, 69 ; of periodic functions: 145, 65; of loots of an equation: 888, 99; see also Series Exponential function : 23, 11 Fourier: 170, 76 Fredholm : 213, ftn Fuchs : 57, ex 15 Fuchsian transformation : 57, ex 15 Functions, algebraic : 233, 98 , 240, 100; analytic: see Analytic func- tions and Analytic functions of sev- eral variables; analytic except for poles: 90, 41, 101, 48, 1S6, 61; branches of : 15, 6 , 29, 13 ; class of integial: 132, 58; of a complex variable: 6, 2; continuous: 6, 2; defined by differential equations: 208, 86; dominant: 56, ex. 7, 81, 35, 227, 94; doubly periodic: 145, 65, .Z40, 67; elementary transcen- dental : 18, 8 ; elliptic : see Elliptic functions; even and odd: 153, Notes; exponential: 23, 11; Gamma: 100, 47 , 229, 96; holomorphic: 11, ftn ; implicit : 233, 98 ; integral : see Integial functions and Integral transcendental functions; inveise, of the elliptic integral : 172, 77 ; in- verse sine: 114, 54; mveise trigo- nometric : 30, 14 ; irrational : 13, 6; logarithms: 28, 13; meromor- phic: 90, ftn ; monodromic: 17, ftn ; monogenic : 9, ftn ; multiform : 17, ftn ; multiple-valued : 17, 7 ; p(u) : 154, 69 ; periods of : 145, 65 , 152, 68 , 178, 77 , 184, 79 ; primary (Weierstrass's) : 127, 57 ; pumitive : 3, 15; rational: 12, 5, 33, 15; rational, of sm z and cos 2 . 85, 16 ; regular in a neighborhood . 0, 40 ; regular at a point : 88, 40 ; regular at the point at infinity : 109, 52 ; represented by definite integrals: 227, 95; senes of analytic: 86, 39; <r (u) : 152, 72 ; single-valued : see Single-valued functions and Single- valued analytic functions; 0(w): 170, 76; trigonometric: 26, 12; f (M) : 15P, 71 ; see ateo Expansions Fundamental formula of the integral calculus : 63, 26 , 72, 31 Fundamental theorem of algebra: 104, Note Gamma function : 1 00, 47 , 229, 96 Gauss : 125, ex. 21 Gauss's sums : 125, ex 21 General linear transformation: 44, ex 2 Geographic maps : see Maps Gouner : 126, ex 28 Goursat: 208, ftn.; #.?, ftn Goursat's theorem : 69, 29 and ftn. 256 INDEX Hadamatd: 206 ', ftn , 212, 88, 2 '18, ex 8 Hermite : 106, 51 , 100, ex , 165, 73 , Jf&S, 75 , 105, ex 9 , 215, 90 and ftn , 216, ftn , j?7, exs 4, 5, 6 , 230, ftn Hermite's formula : 215, 90 ; for ellip- tic integrals: 165, 73 , 168, 75 , .Z05, ex 9 Holomorphic functions : 11, ftn Hyperbolic tiansformations: 57, ex 15 Hypei elliptic integrals. 116, 55, #47, 103; periods of : 116, 55 Imagmaries, conjugate : 4, 1 Imaginary quantity : 8, 1 Implicit functions, Weierstrass's theo- rem : 233, 98 , see also Functions, inverse, and Lagrange's formula Independent periods, Jacobi's theo- rem : 147, 66 Index of a quotient : 103, 49 Infinite number, of singular points; 134, 60, see also Mittag-Leffler's theorem 5 of zeros: 26, 11 , 93, 42 , 128, 57 , see also Weierstrass's theo- rem Infinite products: 22, 10, 129, 57, 194, exs. 2 and 3 ; uniform conver- gence of, 22, 10 , 129, 37 , see also Expansions Infinite series : see Series Infinity : see Point at infinity Inflection, point of : 186, 80 Integral functions: 21, 8, 127, 57; associated: 218, ex. 7; class of: 132, 58 ; with an infinite number of zeros: 127, 57; periodic: 147, 65; transcendental : 21, ftn , 00, 42 , JJff, 61 , 230, 96 Integral transcendental functions : 21, ftn., ft*, 42 , 136, 61 , 230, 96 Integrals, Abelian : 193, 82 , #4#, 101 ; Abelian, of the first, second, and third kind: 244, 101; Abel's theo- rem: 244, 102; Cauchy's: 76, 33, change of variables in: 62, 26; along a closed curve: ##, 28; definite: see Definite integrals; diffeientia- tion of. 77, 33, 227, 95; double: see Double integrals; elliptic: 120, 56 , 174, 78 , 250, 103 ; of elliptic functions: 168, 75; fundamental formula of the integral calculus: 63, 26 , 72, 31 ; Hermite's formula : 215, 90; Hei mite's formula foi el- liptic: 165,1%, 168,75, 195, sx 9, hyperelhptic : 116, 55, 247, 103; law of the mean (Weierstrass, Dar- boux): 64,27, line: &Z, 25, 62, 26, 74, 32, 224, 93; of rational func- tions: 33, 15, -WS, 53; of senes: 86, 39; uniform convergence of: 229, 96 ; see also Cauchy's theorems Invariants (integrals) . 57, ex 15 ; of elliptic functions : 158, 70 , 172, 77 , 188, 79 Inverse functions: see Functions, in- verse, implicit Inversion : 45, 19 , 57, exs 13 and 14 Irrational functions: 13, 6, see also Functions Irreducible polynomial : 240, 100 "* Isolated singular points: 89, 40 , 132, 59 ; essentially singular : 91, 42 Isothermal curves : 54, 24 Jacdbi : 125, ex 18 , 147, 66 , 154, 69 , 170, 76 , 180, 78 Jacobi's theorem : 147, 66 Jensen : 104, 50 Jensen's formula : 104, 50 Kepler: 109, ex , jf#0, ex 26 Kepler' s equation : 109, ex , 126, ex 27 JTfem : 0, ex 23 Lagrange : ./0, 51 , 1##, ex. 26 , 251, 104 Lagrange's formula: 106, 51, .?<?, ex. 26 ; extension of : 250, 104 Laplace: 10, 3, 54, 24, 55, Note, 106, 51 , j?5, ex 19 , 251, 104 Laplace's equation : 10, 3 , 54, 24 , 55, Note Laurent: 75, 33, AT, 37, 91, 42, 04, 43, JW, ex.23, 146,6$ INDEX 257 Laurent's series: 75,33, SI, 37, 146, 65 Law of the mean for integrals . 64, 27 Legendre : 106, ex , 125, ex 18 , 180, 78 Legendre's polynomials: 108, ex ; Jacobi's f orm: 125, ex 18; Laplace's form : 125, ex 19 Limit point : 90, 41 Line integrals: 61, 25, 62, 26, 74, 32 , 84, 93 Linear tiansformation : 59, ex 23; general : 44, ex 2 Lines, singular: see Natural bound- aries, and Cuts Lwumlle: <W, 36, 150,51 Liouville's theorem : 81, 36 , 150, 67 Logarithmic critical points : 32, 14 , 118, 53 Logarithms : 28, 13 , H5, 53 ; natural or Napieuan: 28, 13; series for Log (1 + ) : 38, 17 Loops: 118, 53, 115, 54, #44, 101 Loxodromic curves : 53, ex 1 Maclaunn: 83, ex Maps, conf oimal : 42, 19 , 45, 20 , 48, 20 , 52, 23 ; geographic: 5#, 23 , see also Projection Meray : 81, ftn. , 200, f tn Mercator's projection . 52, ex 1 Meromorphic functions : 90, f tn. Mtitag-Leffler: 127, 57 and ftn , J734, 61 , 1S9, 63 Mittag-Leffler's theorem : 127, 57 , 1S4, 61 , 155, 63 ; Cauchy's method : 139, 63 Monodromic functions : 17, ftn. Monogemc functions : 9, ftn. Mor&ra : 7S, 34 Mor era's theorem : 78, 34 Multiform functions: 17, ftn Multiple-valued functions : 17,1 Napier i 28, 13 Napierian logarithms : 28, 13 Natural boundary : 201, 84 , 208, 87 , 211, 88 Natural logarithms : 28, 13 Neighborhood: &, 40; of the point at infinity: 109, 52 Odd functions : 154, 68 Order, of elliptic functions : 150, 68 ; of poles: 89, 40; of zeros: 88, 40 Ordinary point : 88, 40 P function, p(u): 154, 68, 182, 79; defined by invariants: 182, 79; le- lation between p (u) and p' (u) : 15#, 70 Pamleve*: 85, 38 Parabolic transformation : 57, ex. 15 Parallelogram of periods : 150, 67 Parametnc representation: see Curves Periodic functions . 14 5, 65 ; doubly : 145, 65, 149, 67, see also Elliptic functions Penodic integral functions : 147, 65 Periods : of ctnx : 144, Note 3; cyclic : 244, 101 ; of definite integrals: 112, 53 , 1 14, Note ; of elliptic functions : 158, 68 , 172, 77, 184, 79; of elliptic integrals: 180, 56; of functions: 145, 65 ; of hyperelhptic integrals : 116, 55; independent: 147, 66; parallelogram of: 150, 67; polar: 118, 53 , 119, 55 , #44, 101 ; primi- tive pair of: 149, ftn ; i elation be- tween periods and invariants: 172, 11 ; of sin x : 143, Note 1 Ptcardi 21, ftn., 93, 42 , 127, ftn. Poincarfi 208, ftn , , ftn., #&, ftn Point, critical or branch : see Critical points; double: 184, 80, 191, 82; at infinity: 109, 52; of inflection: 186, 80; limit: 90, 41; ordinary: 88, 40; symmetnc: 5, ex 17, see also Neighborhood, Singular points, and Zeros Polar periods : see Periods, polar Poles: 88, 40, 90, 41, 133, 59; of elliptic functions: 150, 68, 154, 68; infinite number of: 135, 61; 137, 62 ; at infinity : 110, 52 ; order of: 0,40 Polynomials, irreducible : 240, 100 258 INDEX Power series : 18, 8 , 196, 83 ; con- tinuity of : 7, 2 , 56, ex 7 ; deriva- tive of : 19, 8 ; dominating C?l, 9 ; lepresentmg an analytic function: 20, 8 , see also Analytic extension, Circle of convergence, and Senes Pi imary functions, Weieisti ass's : 127, 57 Primitive functions : S3, 15 Primitive pan of penods : 149, ftn Pimcipal part: 89, 40, 91, 42, 110, 52 , 133, 59 , 135, 61 Principal value, of arc sm z: SI, ftn Products, infinite : see Infinite products Projection, Mercator's: 52, ex 1; stereographic : 53, ex 2 Puiseux : 240, ftn Quantity, imaginary or complex : 3, 1 Quartics: 187, 81; bicncular: 193, ex Rational fraction : 1S3, 59 Rational functions: 12, 5; integrals of : 33, 15 ; of sin z and cos z : 35, 16 Region, connected : 11, 4 Regular functions: see Functions, regular Representation, conformal: see Con- formal repiesentation; parametric: see Curves Residues: 75, 33, 94, 43, 101, 48, 110, 62, 112, 53; ot elliptic func- tions: 151, 68; sum of: 111, 52; total : 111, 52 Rhumb lines : 53, ex. 1 Riemann : 10, ftn , 50, 22 ; 74, 32 , #44, 101 Riemann surfaces : 244, 101 Riemann's theorem : 50, 22 Roots of equations: see Equations, D'Alembert's theorem, and Zeros Sauvage : 2S1, 97 Schroder : #14, 89 Senes, of analytic functions : 86, 39 ; AppelPs: 84, 38; Burman's: l#tf, ex 26; the Caucby-Laurent : 81, 35; the Cauchy-Taylor : 79, 35 ; for ctn x: 143, 64; differentiation of: 88, 39; dominant: SI, 9, 157, 69; double : see Double senes , integia- tion of : 86, 39 ; Laurent's : 75, 33 , 81, 37, 146, 65; for Log (1 + z) : 38, 17 ; of polynomials (Pamleve*) : 86, 38; for tan z, etc : 154, ex 4; Taylor's: SO, 8, 75, 33, 78, 35, #0#, ftn , 826, 94; umfoimly con- vergent : 7, 2 , 86, 39 , 55, 39 , see also Lagrange's formula, Mittag- Leffler's theorem, and Power series Seveial variables, functions of. 218, 91 , see also Analytic functions of several variables Sigma function, <r(u) : 162, 72 Single-valued analytic functions : 127, 57; with an infinite number of singular points, Mittag-Leffler's the- orem: 1^4,60; (Cauchy's method) : 139, 63; with an infinite number of zeros, Weierstrass's theoiem: 128, 57; primary functions: 107, 57 Single-valued functions: 17, 7, 127, 57 Singular lines: see Cuts and Natuial boundaries Singular points: 13, 5, 75, 33, 88, 40, 204, 85, 232, 97; algebraic: 241, 100 ; on circle of convergence : 202, 84 and ftn; essentially: 91, 42; essentially, at infinity: 110, 52; infinite number of: 134, 60, 139, 63; isolated: 89, 40, 132, 59; log- arithmic: #44, 101; order of: 89, 40 ; transcendental : 248, Note , see also Critical points, Mittag-Leffler's theorem, and Poles Singularities of analytical expressions: 213, 89 , see also Cuts Stereographic projection : 53, ex. 2 StteUjes: 10$ ex. Symmetric points : 58, ex. 17 Systems^ conjugate isothermal : 54, 24 Tannery : 214, 89 Taykr: 20, 8, 75, 33, 78, 35, 206, ftn., 226, 94 INDEX 259 Taylor's formula, series : 20, 8 , 75, SB , 78, 35 , 806, ftn ; for double series : 226, 94 Theta function, 9 (u) : 170, 76 Total lesidue: 111, 52 Tianscendental functions: see Func- tions Transcendental integral functions see Integral tianscendental functions Transformations, bnational: 192, 82, 252, ex 1; circular: 45, 19, 57, ex 13, conformal. 42, 19, 45, 20, 48, 20, 52, 23; elliptic: 57, ex. 15; Fuchsian: 57, ex 15; general hn- eai : 44, ex 2 ; hypeibolic : 57, ex 15 ; mveision : 45, 19 , 57, exs 13 and 14 ; linear : 59, ex 23 ; parabolic . 57, ex 15 , see also Projection Tiigonometnc functions: 26, 12; in- verse : SO, 14 ; inverse sine . 11 4, 54, penod of ctn x: 144, Note 3; peiiod of sin x : 148, Note 1 ; prin- cipal value of: 31, ftn ; lational functions of sin z and cos z : 35, 16 , see also Expansion Umcuisal cuives: 191, 82 Uniform convergence: see Conver- gence, uniform Uniform functions : 17, ftn Unifoimly conveigent senes and prod- ucts : see Conveigence, uniform Variables, complex: 6, 2; infinite values of* 109, 52; several: see Analytic functions of several vari- ables Wezerstrass*. 64, 27, 88, ftn., 92, 42, 121, 56 , 187, 57 and ftn , 139, 63 , 149, 67 , 154, 69 , 156, 69 , 800, ftn , 212, 88 , 283, 98 , 237, ftn Weierstrass's formula: 64, 27, 131, 56; primary functions: 7^7, 57; theorem : 92, 42 , 127, 57 ; J?#, 62 , 139, 63 , 233, 98 Zeros, of analytic functions : 88, 40 , 884, 98 , #4.T, 100 ; of elliptic func- tions : 152, 68 , 154, 68 ; infinite number of : 26, 11 , 93, 42 , 188, 57; order of: , 40, see also D'Alembert's theorem Zeta function, f (u) : 159, 71 Date Due Demco 293-5 3 fiHfl2 D1D37 filll Carnegie Institute of Technology Library Pittsburgh, Pa. 8340 o o